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Classroom Tips and Techniques: Eigenvalue Problems for ODEs - Part 3 Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Initializations</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9 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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Introduction</Font></Text-field> In Part 1 of this series of articles on solving eigenvalue problems for ODEs, we discussed equations for which the general solution readily yielded eigenvalues and eigenfunctions without the need for detailed knowledge of any of the special functions of applied mathematics. In Part 2 of this series, we examined the solution of Laplace's equation in a cylinder. Separation of variables in cylindrical coordinates leads to a singular Sturm-Liouville eigenvalue problem whose differential equation is the Bessel equation. In Part 3 of this series, we will examine the solution of Laplace's equation in a sphere. Separation of variables in spherical coordinates leads to a singular Sturm-Liouville eigenvalue problem in which the differential equation is Legendre's equation. Reasoning from a general solution of Legendre's equation to the bounded solutions needed to solve the eigenvalue problem is a significantly greater challenge than it was for the parallel case of Bessel's equation. Our discussion will highlight the contributions Maple can make to this process.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Steady-State Temperatures in a Sphere</Font></Text-field> At steady state, the temperature LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in a sphere satisfies Laplace's 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 and some conditions on the boundary of the sphere, which we describe in spherical coordinates by 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 In addition to the conditions prescribed on the surface LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJjRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRjY2LVEiLEYnRjJGOS9GPEZQRj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS1EsMC4zMzMzMzMzZW1GJ0Yythe physical properties of the system demand the solution be continuous. This requirement will be the most important, and most difficult condition to impose. If the temperature on the surface of the sphere is prescribed, we say that a Dirichlet condition has been imposed. If the prescribed temperature on the surface is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY2O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj8tSSNtb0dGJDYtUSIsRidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRj8=a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy alone, the temperature in the sphere will exhibit azimuthal symmetry so that 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 Alternatively, if this prescribed temperature is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTY2O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RidGPUY/Rj9GP0ZBRj8=then the temperature in the sphere will exhibit azimuthal asymmetry so that 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 If the surface of the sphere is insulated so the net heat flux across this surface is zero, we say that a homogeneous Neumann condition has been imposed. The flux across the surface is the normal derivative given by 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 evaluated at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJjRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRjY2LVEiLEYnRjJGOS9GPEZQRj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS1EsMC4zMzMzMzMzZW1GJ0Yywhere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the radius of the sphere. The net flux would be the surface integral of this derivative. However, if the net heat flux across the surface of a homogeneous sphere is zero, the steady-state temperature in the sphere will be constant.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Spherical Coordinates in Maple</Font></Text-field> From our statement of the problem above, our definition of spherical coordinates can be inferred. However, because there are two different usages prevalent in the literature, we will explicitly define our system according to the notation in most mathematics texts. In such texts, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is the distance from the origin; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjJGOS9GPEYxRj5GQEZCRkRGRkZIL0ZMRkpGMg==measured from the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis and around the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, lies in the range LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHRjhGL0YyLUYsNiRRIjJGJ0YvLUYzNi1RIn5GJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRJjAuMGVtRicvRklGVy1GSzYlUScmIzk2MDtGJ0ZORi9GLw==; and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMg==measured downward from the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, lies in the range LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHRjhGL0YyLUZLNiVRJyYjOTYwO0YnRk5GLy1GMzYtUSIuRidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSYwLjBlbUYnL0ZJRldGLw== The equations connecting these spherical coordinates with Cartesian coordinates appear on the left in Table 1. Spherical coordinates in texts for physics, engineering, and the applied sciences tend to interchange the names LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYy The equations connecting these spherical coordinates with Cartesian coordinates appear on the right in Table 1. Math Texts Angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy measured down from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis Science Texts Angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy measured down from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis 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 Table 1 Spherical coordinates as defined in math texts (left) and science texts (right)
Finally, note that in Maple's VectorCalculus package, commands that use spherical coordinates assume that the "middle" coordinate in the triple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidkYnRi9GMkY1LUYsNiVRIndGJ0YvRjJGOQ== is the angle measured down from the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis. Unfortunately, in a number of plot commands in the plots package, this convention is not respected. Maple is currently struggling with this quandary, especially so, given its commitment to backward compatibility. We set the ambient coordinate system via the command 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
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Laplace's Equation in Spherical Coordinates</Font></Text-field> In a sphere, the steady-state temperature 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 satisfied Laplace's equation 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 This equation is given in Maple as 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMRUdGKC8sLCooIiIjIiIiSSRyaG9HRighIiItSSVkaWZmR0YmNiQtSSJ1R0YoNiVGM0kpJnZhcnBoaTtHRihJJnRoZXRhR0YoRjNGMkYyLUY2NiRGOC1JIiRHRiY2JEYzRjFGMioqKUYzRjFGNC1JJHNpbkdGJTYjRjtGNC1JJGNvc0dGJUZGRjItRjY2JEY4RjtGMkYyKiZGQ0Y0LUY2NiRGOC1GQDYkRjtGMUYyRjIqKEZDRjQpRkRGMUY0LUY2NiRGOC1GQDYkRjxGMUYyRjIiIiE3Iy8sLEYwRjItRjY2JEY1RjNGMkZCRjIqJkZDRjQtRjY2JEZJRjtGMkYyKihGQ0Y0RlFGNC1GNjYkLUY2NiRGOEY8RjxGMkYyRlY= Maple can determine if the partial differential equation is variable separable: 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 IiIh The return of "0" indicates that the equation is indeed separable because the separability conditions are identically satisfied.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Azimuthal Symmetry</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Separation of Variables</Font></Text-field> Under the assumption that the steady-state temperatures LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are symmetric about the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, dependence on angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy can be dispensed with. Hence, 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and a Maple-generated variable-separation is obtained with 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSScmd2hlcmVHRig2JC8tSSJ1R0YoNiRJJHJob0dGKEkpJnZhcnBoaTtHRigqJi1JJF9GMUdGKDYjRjMiIiItSSRfRjJHRig2I0Y0Rjk3IzwkLy1JJWRpZmZHRiY2JEY2LUkiJEdGJjYkRjMiIiMsJiooKUYzRkYhIiJGNkY5JkkjX2NHRiU2I0Y5RjlGOSooRkZGOUYzRkotRkE2JEY2RjNGOUZKLy1GQTYkRjotRkQ2JEY0RkYsJiomRjpGOUZLRjlGSiooLUkkc2luR0YlRjxGSi1JJGNvc0dGJUY8RjktRkE2JEY6RjRGOUZKNyMtSSxQREVTb2xTdHJ1Y0dGJjYkRi83IzwkLy1GQTYkRmduRjRGVi8tRkE2JEZPRjNGRw== Equation 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 shows that a variable separation solution of the form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJVR0YoKiYtSSRSaG9HRig2I0kkcmhvR0YoIiIiLUkkUGhpR0YoNiNJKSZ2YXJwaGk7R0YoRjM3I0Yu exists, and provides the ordinary differential equations the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Mjk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTYxO0YnRi9GMkYyRjJGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY2O0YnRi9GMkYyRjJGMg== must satisfy. We now proceed to obtain these same results from first principles. Under the separation assumption, Laplace's equation assumes the simpler form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIiLywqKioiIiNGMEkkcmhvR0YoRjAtSSRSaG9HRig2I0Y1ISIiLUklZGlmZkdGJjYkRjZGNUYwRjAqKClGNUY0RjBGNkY5LUY7NiRGNi1JIiRHRiY2JEY1RjRGMEYwKiotSSRQaGlHRig2I0kpJnZhcnBoaTtHRihGOS1JJHNpbkdGJUZHRjktSSRjb3NHRiVGR0YwLUY7NiRGRUZIRjBGMComRkVGOS1GOzYkRkUtRkI2JEZIRjRGMEYwIiIhNyMvLCpGM0YwKihGPkYwRjZGOS1GOzYkRjpGNUYwRjBGREYwKiZGRUY5LUY7NiRGTUZIRjBGMEZU Moving all terms in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy to the right, we then have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSV0ZW1wRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYlLUYsNiVRJ3NlbGVjdEYnRjRGNy1GOzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGZ24tSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSRoYXNGJ0Y0RjctRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZmbi9GU1EsMC4zMzMzMzMzZW1GJy1GIzYlLUYsNiVRJGxoc0YnRjRGN0ZZLUZqbjYkLUYjNiMtSSVtc3ViR0YkNiUtRiw2JVEicUYnRjRGNy1GIzYjLUkjbW5HRiQ2JFEiMUYnRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+RmFvLUYsNiVRKSZ2YXJwaGk7RicvRjVGQkY+Rj5GKy1GOzYtUSI6RidGPkZARkNGRUZHRklGS0ZNRk9GUg== 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIjLywmKipGMCIiIkkkcmhvR0YoRjQtSSRSaG9HRig2I0Y1ISIiLUklZGlmZkdGJjYkRjZGNUY0RjQqKClGNUYwRjRGNkY5LUY7NiRGNi1JIiRHRiY2JEY1RjBGNEY0LCYqKi1JJFBoaUdGKDYjSSkmdmFycGhpO0dGKEY5LUkkc2luR0YlRkhGOS1JJGNvc0dGJUZIRjQtRjs2JEZGRklGNEY5KiZGRkY5LUY7NiRGRi1GQjYkRklGMEY0Rjk3Iy8sJkYzRjQqKEY+RjRGNkY5LUY7NiRGOkY1RjRGNCwmRkVGOSomRkZGOS1GOzYkRk5GSUY0Rjk= Introduction of Bernoulli's separation constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy then leads to the ordinary differential equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Jy1JJW1zdWJHRiQ2JS1GLDYlUSJxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSNtbkdGJDYkUSIzRicvRjtRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSomY29sb25lcTtGJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduLUYjNiUtRiw2JVEkbGhzRidGN0Y6LUZJNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkNGTEZPRlFGU0ZVRldGWS9GZm5RJjAuMGVtRicvRmluRmNvLUkobWZlbmNlZEdGJDYkLUYjNiMtRjI2JUY0LUYjNiMtRkA2JFEiMkYnRkNGRUZDLUZJNi1RIj1GJ0ZDRkxGT0ZRRlNGVUZXRllGZW5GaG4tRiw2JVEpJmxhbWJkYTtGJy9GOEZORkNGKw== 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIkLywmKioiIiMiIiJJJHJob0dGKEY1LUkkUmhvR0YoNiNGNiEiIi1JJWRpZmZHRiY2JEY3RjZGNUY1KigpRjZGNEY1RjdGOi1GPDYkRjctSSIkR0YmNiRGNkY0RjVGNUknbGFtYmRhR0YoNyMvLCZGM0Y1KihGP0Y1RjdGOi1GPDYkRjtGNkY1RjVGRQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIlLywmKiotSSRQaGlHRig2I0kpJnZhcnBoaTtHRighIiItSSRzaW5HRiVGNkY4LUkkY29zR0YlRjYiIiItSSVkaWZmR0YmNiRGNEY3Rj1GOComRjRGOC1GPzYkRjQtSSIkR0YmNiRGNyIiI0Y9RjhJJ2xhbWJkYUdGKDcjLywmRjNGOComRjRGOC1GPzYkRj5GN0Y9RjhGSA== We are primarily interested in the second of these equations - it will become Legendre's equation after a mild rearrangement and change of variables. First, write the equation in the form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiImLyIiISwoSSdsYW1iZGFHRigiIiIqKi1JJFBoaUdGKDYjSSkmdmFycGhpO0dGKCEiIi1JJHNpbkdGJUY5RjstSSRjb3NHRiVGOUY1LUklZGlmZkdGJjYkRjdGOkY1RjUqJkY3RjstRkE2JEY3LUkiJEdGJjYkRjoiIiNGNUY1NyMvRjIsKEY0RjVGNkY1KiZGN0Y7LUZBNiRGQEY6RjVGNQ== and then 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiInLyIiISwoKigtSSRQaGlHRig2I0kpJnZhcnBoaTtHRigiIiItSSRzaW5HRiVGN0Y5SSdsYW1iZGFHRihGOUY5KiYtSSRjb3NHRiVGN0Y5LUklZGlmZkdGJjYkRjVGOEY5RjkqJkY6RjktRkE2JEY1LUkiJEdGJjYkRjgiIiNGOUY5NyMvRjIsKEY0RjlGPUY5KiZGOkY5LUZBNiRGQEY4RjlGOQ== Now, make the change of variables 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with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5MzQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY2O0YnRi9GMkYyRjItRiw2I1EhRidGMg== becoming 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 This is done in Maple with 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIoLyIiISomLDAqJi1JInlHRig2I0kieEdGKCIiIkknbGFtYmRhR0YoRjpGOiooRjZGOkY7RjopRjkiIiNGOiEiIiooRj5GOi1JJWRpZmZHRiY2JEY2RjlGOkY5RjpGPyooRj5GOilGOSIiJEY6RkFGOkY6LUZCNiRGNi1JIiRHRiY2JEY5Rj5GOiooRj5GOkZHRjpGPUY6Rj8qJkZHRjopRjkiIiVGOkY6RjopLCZGOkY6KiRGPUY6Rj8jRjpGPkY/NyMvRjIqJiwwRjVGOkY8Rj9GQEY/RkRGOi1GQjYkRkFGOUY6KihGPkY6RlhGOkY9RjpGPyomRlhGOkZORjpGOkY6RlBGPw== Further simplifying, we have 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIpLyIiISwoKiYsJiIiIkY2KiQpSSJ4R0YoIiIjRjYhIiJGNi1JJWRpZmZHRiY2JC1JInlHRig2I0Y5LUkiJEdGJjYkRjlGOkY2RjYqKEY6RjYtRj02JEY/RjlGNkY5RjZGOyomRj9GNkknbGFtYmRhR0YoRjZGNjcjL0YyLCgqJkY1RjYtRj02JEZGRjlGNkY2RkVGO0ZIRjY= which is the standard form of Legendre's equation, the self-adjoint form of which would be 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">The Sturm-Liouville Eigenvalue Problem</Font></Text-field> The eigenvalue problem that embeds Legendre's equation is singular. The boundary conditions are simply that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== must be continuous on the interval 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 Passage from the general solution 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 Ly1JInlHNiI2I0kieEdGJSwmKiZJJF9DMUdGJSIiIi1JKkxlZ2VuZHJlUEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCYqJiNGKyIiI0YrKSwmKiYiIiVGK0knbGFtYmRhR0YlRitGK0YrRitGNEYrRitGNCEiIkYnRitGKyomSSRfQzJHRiVGKy1JKkxlZ2VuZHJlUUdGLkYxRitGKw== to the eigenfunctions is surprisingly more difficult than it was for Bessel's equation. Because we are in extended typesetting mode, the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== are displayed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImFGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj1GPUY9 and 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respectively. (Were we in extended typesetting mode during our earlier discussion of Bessel's equation, Maple would have displayed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoQmVzc2VsSkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUSIsRidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0Y9Rj9GMkYyRjI= as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImFGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj1GPS1JI21vR0YkNi1RIi5GJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZQLyUpc3RyZXRjaHlHRlAvJSpzeW1tZXRyaWNHRlAvJShsYXJnZW9wR0ZQLyUubW92YWJsZWxpbWl0c0dGUC8lJ2FjY2VudEdGUC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmluRj0=) When solving Laplace's equation in the cylinder, it was relatively easy to use continuity to restrict the general solution to just 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, the Bessel function bounded on the interval 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 to determine the eigenvalues LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRImtGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== from the zeros of 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 We began the process by ruling out the Bessel function of the second kind because we could tell from a graph that all such functions were unbounded at the origin. We will try to rule out the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== in a similar way, but we will find the process more difficult than it was for the Bessel function. For example, consider 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 LCwqJiMiIiIiIiVGJS1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkkieEdGLEYlRiVGJUYlISIiKiZGJEYlLUYoNiMsJkYlRjBGL0YlRiVGJSooIyIiJEYmRiUpRi8iIiNGJUYnRiVGJSooRjZGJUY4RiVGMkYlRjAqJiNGN0Y5RiVGL0YlRjA= from which it is clear that the function is unbounded at the endpoints LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSUmcG07RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlMtSSNtbkdGJDYkUSIxRidGOUY5 because of the logarithms. But this is obvious for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=, an integer. It is a bit more difficult to divine the endpoint behavior for general values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. For example, we can calculate the values LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2JS1GLDYlUSpMZWdlbmRyZVFGJy9GN0ZERkBGPC1GVzYkLUYjNiYtSSNtbkdGJDYkUSQyLjVGJ0ZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GIzYkLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZfcC1GYG82JFEiMUYnRkBGK0ZARitGQEYrRis= 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 LCYkIiIiSSlpbmZpbml0eUclKnByb3RlY3RlZEdGJComRiNGJF4jRiRGJEYk LCYkIiIiSSlpbmZpbml0eUclKnByb3RlY3RlZEdGJComRiNGJF4jRiRGJEYk which suggest LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== may indeed be unbounded at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSUmcG07RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlMtSSNtbkdGJDYkUSIxRidGOUY5 for general values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Figure 1 contains graphs of the real and imaginary parts of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGQUYywith LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the open interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIjFGJ0Y0LUYxNi1RIixGJ0Y0RjcvRjtRJXRydWVGJ0Y8Rj5GQEZCRkQvRkdRJjAuMGVtRicvRkpRLDAuMzMzMzMzM2VtRidGS0Y0RjRGNA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ=. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2L0YrLUYjNiUtRiw2JVEnJnJlYWw7RicvRjdGREZARjwtRlc2JC1GIzYlRistRiM2JS1GLDYlUSpMZWdlbmRyZVFGJ0ZqbkZARjwtRlc2JC1GIzYlLUYsNiVRImFGJ0Y2RjktRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0Y2RjlGQEYrRkBGW3AtRiM2JkZoby1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmpwLUYjNiUtSSNtbkdGJDYkUSIxRidGQC1GPTYtUSMuLkYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJ0ZULUZfcTYkUSIzRidGQEYrRltwLUYjNiZGYXBGZnAtRiM2JkYrLUYjNiQtRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPRmVxL0ZVRmZxRl5xRmJxRl5xRitGW3AtRiM2JS1GLDYlUSVheGVzRidGNkY5RmZwLUYsNiVRJGJveEYnRjZGOUZbcC1GIzYlLUYsNiVRLG9yaWVudGF0aW9uRidGNkY5RmZwLUZXNiYtRiM2JS1GX3E2JFEjMzBGJ0ZARltwLUZfcTYkUSM3MEYnRkBGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZbcC1GIzYlLUYsNiVRJnRpdGxlRidGNkY5RmZwLUZXNiYtRiM2JUYrLUkjbXNHRiQ2I1EwKGEpfi1+UmVhbH5QYXJ0RidGK0ZARlt0Rl50RitGQEYrRis= 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Figure 1 Real and imaginary parts of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== for 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suggesting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== is unbounded on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIxRidGLy1GLDYtUSYmbGVxO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZOLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0ZKRkZGLw==
From Figure 1(a) especially, we conclude that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSIxRidGMkYyRjItRiw2I1EhRidGMg== is unbounded for general values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. On the basis of this conclusion, we set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0MyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== to zero in the general solution of Legendre's equation, and turn our attention to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGQUYythe Legendre function of the first kind. We first show that for general (real) values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGOQ==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 is unbounded. Sample calculations include LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjkvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSQyLjJGJ0Y5LUY9Ni1RIixGJ0Y5RkAvRkNRJXRydWVGJ0ZERkZGSEZKRkxGTi9GUlEsMC4zMzMzMzMzZW1GJy1GIzYkLUY9Ni1RKiZ1bWludXMwO0YnRjlGQEZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZjby1GWTYkUSIxRidGOUYrRjlGK0Yr LCYkIiIiSSlpbmZpbml0eUclKnByb3RlY3RlZEdGJComRiNGJF4jRiRGJEYk To illustrate this behavior for multiple values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we define the following piecewise function. 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoLUkqUElFQ0VXSVNFR0YlNiQ3JCIiIzIiJSs1LUkkYWJzR0YmNiMtSSpMZWdlbmRyZVBHRiU2JEkiYUdGKCEiIjckRjhJKm90aGVyd2lzZUdGKDcjLUkqcGllY2V3aXNlR0YmNiVGM0YyRjg= If 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 is large, then a graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== will show a point at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjJGJ0Y+Rj5GPkY+ for that value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. If 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 is "not large" then a graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== will show the value of 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 We can control the evaluation points for a graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== if we define the uniform random variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= via LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJYRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYlLUYsNiVRL1JhbmRvbVZhcmlhYmxlRidGNEY3LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEmMC4wZW1GJy9GU0Znbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2JS1GLDYlUShVbmlmb3JtRidGNEY3RlktRmpuNiQtRiM2JkYrLUYjNiQtRjs2LVEqJnVtaW51czA7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRl1wLUkjbW5HRiQ2JFEiM0YnRj4tRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZmbi9GU1EsMC4zMzMzMzMzZW1GJ0ZfcEY+RitGPkYrLUY7Ni1RIjpGJ0Y+RkBGQ0ZFRkdGSUZLRk1GT0ZS then create a uniform but random sample of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-values that includes the integers in the interval 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. 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 The graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== in Figure 2 shows that virtually all evaluations of 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 are large in magnitude. 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 NiktJSdDVVJWRVNHNiM3Z3E3JCQhI0khIiIkIiM1ISIiNyQkIStiWT5zSCEiKiQiIz8hIiI3JCQhKyRlKGVHSCEiKiQiIz8hIiI3JCQhKzxOSyR6IyEiKiQiIz8hIiI3JCQhKylHKkh3RSEiKiQiIz8hIiI3JCQhK2pVKFthIyEiKiQiIz8hIiI3JCQhK0ZvJTRgIyEiKiQiIz8hIiI3JCQhK0s8MihcIyEiKiQiIz8hIiI3JCQhK0ZcUSRcIyEiKiQiIz8hIiI3JCQhK3pMM2dCISIqJCIjPyEiIjckJCErNlIsJ0cjISIqJCIjPyEiIjckJCErXEZjP0EhIiokIiM/ISIiNyQkIStbJyllJD0jISIqJCIjPyEiIjckJCErVkxEb0AhIiokIiM/ISIiNyQkISs9aHdFQCEiKiQiIz8hIiI3JCQhK25mQi9AISIqJCIjPyEiIjckJCErJyk9dCYzIyEiKiQiIz8hIiI3JCQhK15oIXAtIyEiKiQiIz8hIiI3JCQhKyplSFYtIyEiKiQiIz8hIiI3JCQhK0J3NTE/ISIqJCIjPyEiIjckJCEjPyEiIiQhIzUhIiI3JCQhKiR5IiozPiEiKSQiIz8hIiI3JCQhK3RWd3k9ISIqJCIjPyEiIjckJCErKFxHLyM9ISIqJCIjPyEiIjckJCEraiRHcmwiISIqJCIjPyEiIjckJCErKT1RaGkiISIqJCIjPyEiIjckJCErKT1dKVE6ISIqJCIjPyEiIjckJCErX29cJFwiISIqJCIjPyEiIjckJCErRXBJdTkhIiokIiM/ISIiNyQkISsyJEglcDkhIiokIiM/ISIiNyQkISt2LyZcWCIhIiokIiM/ISIiNyQkISskZXgyVyIhIiokIiM/ISIiNyQkISskSHNAVSIhIiokIiM/ISIiNyQkISssRD08OSEiKiQiIz8hIiI3JCQhK1EmXFFNIiEiKiQiIz8hIiI3JCQhKygpZSdcRyIhIiokIiM/ISIiNyQkIStbKDRGOCIhIiokIiM/ISIiNyQkISo3LnU0IiEiKSQiIz8hIiI3JCQhIzUhIiIkIiM1ISIiNyQkIStPOGtzKCohIzUkIiM/ISIiNyQkISlrJm9kKiEiKSQiIz8hIiI3JCQhK1RTKDQrKiEjNSQiIz8hIiI3JCQhK3cmSC8hKikhIzUkIiM/ISIiNyQkISsiPVxLPCghIzUkIiM/ISIiNyQkISt1RFwxciEjNSQiIz8hIiI3JCQhK2E1LzhnISM1JCIjPyEiIjckJCErXy48OlQhIzUkIiM/ISIiNyQkIStAJVFgbiQhIzUkIiM/ISIiNyQkISs4UUlSTSEjNSQiIz8hIiI3JCQhK2R6I1tFJCEjNSQiIz8hIiI3JCQhKjQvdidIISIqJCIjPyEiIjckJCErUDpjTz0hIzUkIiM/ISIiNyQkISttIXBFZyIhIzUkIiM/ISIiNyQkIStGRE9UaCEjNiQiIz8hIiI3JCQhKzIiKW86KSohIzckIiM/ISIiNyQkIiIhISIiJCIjNSEiIjckJCIrKio0QnVOISM2JCIjPyEiIjckJCIqOCkpPnIiISIqJCIjPyEiIjckJCIrIj1geSU9ISM1JCIjPyEiIjckJCIrO2hhK0IhIzUkIiM/ISIiNyQkIispekpIJEchIzUkIiM/ISIiNyQkIisoXDtNKUghIzUkIiM/ISIiNyQkIitXJSlIcFMhIzUkIiM/ISIiNyQkIitfJz4lSFQhIzUkIiM/ISIiNyQkIitwYSVlNiYhIzUkIiM/ISIiNyQkIitmXTE7XiEjNSQiIz8hIiI3JCQiKyVbOyo9aCEjNSQiIz8hIiI3JCQiK3Awb2lwISM1JCIjPyEiIjckJCIrMjF5eSgpISM1JCIjPyEiIjckJCIrNGZ1VyMqISM1JCIjPyEiIjckJCIrUS0pZUkqISM1JCIjPyEiIjckJCIjNSEiIiQhIzUhIiI3JCQiK2hnQHk1ISIqJCIjPyEiIjckJCIrPnFHTjYhIiokIiM/ISIiNyQkIitPLlklPiIhIiokIiM/ISIiNyQkIismKSopPWM3ISIqJCIjPyEiIjckJCIrZCY0KilbIiEiKiQiIz8hIiI3JCQiK2NCZzI6ISIqJCIjPyEiIjckJCIrbVhQQTohIiokIiM/ISIiNyQkIisjND8iRzohIiokIiM/ISIiNyQkIit2OD9WOiEiKiQiIz8hIiI3JCQiK0gyNSRmIiEiKiQiIz8hIiI3JCQiKyl5aSVcOyEiKiQiIz8hIiI3JCQiKyJRLl1uIiEiKiQiIz8hIiI3JCQiK1dzcWw8ISIqJCIjPyEiIjckJCIrMiUqPnI8ISIqJCIjPyEiIjckJCIrIikzVCs9ISIqJCIjPyEiIjckJCIrYyozZCk9ISIqJCIjPyEiIjckJCIrQyQ+USE+ISIqJCIjPyEiIjckJCIrbD0hXCY+ISIqJCIjPyEiIjckJCIrbjwoXCk+ISIqJCIjPyEiIjckJCIjPyEiIiQiIzUhIiI3JCQiK09OSVc/ISIqJCIjPyEiIjckJCIrSyNvQEAjISIqJCIjPyEiIjckJCIrWUt2OkEhIiokIiM/ISIiNyQkIis6Jj5hTSMhIiokIiM/ISIiNyQkIituYilRWSMhIiokIiM/ISIiNyQkIitwVC0hWyMhIiokIiM/ISIiNyQkIiskKT47LkQhIiokIiM/ISIiNyQkIitSPGV2RCEiKiQiIz8hIiI3JCQiKzBUMS9FISIqJCIjPyEiIjckJCIrJEhLOHEjISIqJCIjPyEiIjckJCIrXiZbZHYjISIqJCIjPyEiIjckJCIrXlBZZUYhIiokIiM/ISIiNyQkIismWylRckYhIiokIiM/ISIiNyQkIikkM28oSCEiKCQiIz8hIiI3JCQiI0khIiIkISM1ISIiLSUlVklFV0c2JDskISNJISIiJCIjSSEiIjskISM/ISIiJCIjPyEiIi0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiM1ISIiLSUmU1RZTEVHNiMlJlBPSU5URy0lKkFYRVNTVFlMRUc2IyUnTk9STUFMRy0lKFNDQUxJTkdHNiMlLlVOQ09OU1RSQUlORURHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkSSIhIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlcVAhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlXVAhIiItJSlDSElMRFJFTkc2Ig== Figure 2 Stylized graph of 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIzRidGLy1GLDYtUSYmbGVxO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZOLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0ZKRkZGLw==
However, it also suggests that 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 for integer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. For noninteger LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImFGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is unbounded so that the bounded solutions of Legendre's equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiOEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvIiIhLCgqJiwmIiIiRjEqJClJInhHRigiIiNGMSEiIkYxLUklZGlmZkdGJjYkLUkieUdGKDYjRjQtSSIkR0YmNiRGNEY1RjFGMSooRjVGMS1GODYkRjpGNEYxRjRGMUY2KiZGOkYxSSdsYW1iZGFHRihGMUYxNyMvRi0sKComRjBGMS1GODYkRkFGNEYxRjFGQEY2RkNGMQ== will be the eigenfunctions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImtGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== with 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 L0kia0c2IiwmKiYjIiIiIiIjRigpLCYqJiIiJUYoSSdsYW1iZGFHRiRGKEYoRihGKEYnRihGKEYnISIi an integer. Hence, the eigenvalues will be 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 LCYqJClJImtHNiIiIiMiIiJGKEYlRig= that is, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRImtGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGOi1GRTYtUSJ+RidGNUZIRkpGTEZORlBGUkZUL0ZXUSYwLjBlbUYnL0ZaRmluLUkobWZlbmNlZEdGJDYkLUYjNiZGOi1GRTYtUSIrRidGNUZIRkpGTEZORlBGUkZUL0ZXUSwwLjIyMjIyMjJlbUYnL0ZaRmRvLUkjbW5HRiQ2JFEiMUYnRjVGNUY1LUZFNi1RIixGJ0Y1RkgvRktGPkZMRk5GUEZSRlRGaG4vRlpRLDAuMzMzMzMzM2VtRidGOkZELUZnbzYkUSIwRidGNUZqb0Zmb0Zqby1GRTYtUSkmaGVsbGlwO0YnRjVGSEZKRkxGTkZQRlJGVC9GV1EsMC4xNjY2NjY3ZW1GJy9GWkZncC1GRTYtUSIuRidGNUZIRkpGTEZORlBGUkZURmhuRmpuRjU= The first few eigenfunctions are 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMEdGKCIiIjcjRi4= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMUdGKEkieEdGKDcjRi4= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQMkdGKCwmIyIiIiIiIyEiIiomIyIiJEYxRjApSSJ4R0YoRjFGMEYwNyNGLg== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQM0dGKCwmKiYjIiImIiIjIiIiKUkieEdGKCIiJEYzRjMqJiNGNkYyRjNGNUYzISIiNyNGLg== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQNEdGKCwoIyIiJCIiKSIiIiomIyIjTkYxRjIpSSJ4R0YoIiIlRjJGMiomIyIjOkY4RjIpRjciIiNGMiEiIjcjRi4= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNQNUdGKCwoKiYjIiNqIiIpIiIiKUkieEdGKCIiJkYzRjMqJiMiI04iIiVGMylGNSIiJEYzISIiKiYjIiM6RjJGM0Y1RjNGMzcjRi4= which are the Legendre polynomials 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normalized so that 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 These polynomials are graphed in Figure 3. 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Figure 3 The Legendre polynomials 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
That the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== reduces to the polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj1GPUY9 for 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 be seen from the following calculations. For noninteger LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, we first obtain the formal power series expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== via 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 then extract the general term in the first series with 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiIqLCQqNiIiIyIiIilJI1BpR0YmI0Y0RjNGNC1JJkdBTU1BR0YlNiMsJkY3RjQqJkY3RjRJImFHRihGNCEiIkY+LUY5NiMsJEY8RjRGPkY9Rj4tSStwb2NoaGFtbWVyR0YnNiQsJEY8Rj5JImtHRihGNC1GQzYkLCZGPEY0RjdGNEZGRjQtSSpmYWN0b3JpYWxHRiY2IywkKiZGM0Y0RkZGNEY0Rj4pIiIlRkZGNClJInhHRihGTUY0RjQ3I0Yx The pochhammer symbol 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 or "rising factorial" for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= complex generalizes to 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 for complex LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYy If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a nonpositive integer, then 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 Making this transformation and setting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= in the general term of the first series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== gives the general coefficient 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM1LCQqOCIiIyIiIilJI1BpR0YmI0Y0RjNGNC1JJkdBTU1BR0YlNiMsJkY3RjQqJkY3RjRJImFHRihGNCEiIkY+LUY5NiMsJEY8RjRGPkY9Rj4tRjk2IywmRjxGPkkia0dGKEY0RjQtRjk2IywkRjxGPkY+LUY5NiMsKEY8RjRGN0Y0RkVGNEY0LUY5NiMsJkY8RjRGN0Y0Rj4tRjk2IywmKiZGM0Y0RkVGNEY0RjRGNEY+KSIiJUZFRjRGNDcjRjE= For large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= this coefficient is asymptotic to 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 LCQqKiMiIiIiIiNGJS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJI1BpR0YqRiVJImFHRixGJUYlSSJrR0YsISIiRi9GMkYy suggesting that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSIxRidGMkYyRjItRiw2I1EhRidGMg== is unbounded since the series under consideration will behave like the harmonic series at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=. We can confirm this behavior by comparing the general coefficient with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR1EmZmFsc2VGJ0Yy for large LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. In the limit we find the ratio tends to 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 LCQqMCIiIyIiIkkjUGlHJSpwcm90ZWN0ZWRHRiUtSSZHQU1NQUc2JEYnSShfc3lzbGliRzYiNiMsJiNGJUYkRiUqJkYvRiVJImFHRixGJSEiIkYyLUYpNiMsJEYwRiVGMkYxRjItRik2IywkRjBGMkYyLUYpNiMsJkYwRiVGL0YlRjJGJQ== which is finite for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= not an integer. To see that for integer LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjJGMg== reduces to a polynomial, examine the recursion formula for its coefficients. This is most efficiently obtained in Maple via 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 Ly1JInVHNiI2IywmSSJrR0YlIiIiIiIjRikqKiwoSSdsYW1iZGFHRiUhIiJGKEYpKiQpRihGKkYpRilGKS1GJDYjRihGKSwmRihGKUYpRilGLkYnRi4= from which it becomes clear that 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 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GLDYlUSJrRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRjY2LVEifkYnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS0ZXLUkobWZlbmNlZEdGJDYkLUYjNiZGTC1GNjYtUSIrRidGMkY5RjtGPUY/RkFGQ0ZFL0ZIUSwwLjIyMjIyMjJlbUYnL0ZLRlxvLUkjbW5HRiQ2JFEiMUYnRjJGMkYyLUY2Ni1RIi5GJ0YyRjlGO0Y9Rj9GQUZDRkVGVkZYRjI= Hence, 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 Therefore, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImtGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieEYnRj1GP0YyRjItRiw2I1EhRidGMg== is a polynomial of degree LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for . 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Orthogonality of the Eigenfunctions </Font></Text-field> The classical proof of the orthogonality of the eigenfunctions of Legendre's equation is based on integration by parts. The self-adjoint form of the equation, namely, 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 is written once for an eigenfunction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and once for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0= The first equation is multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIsRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y9and the second, by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, and the difference of the two products is integrated over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIxRidGLy1GLDYtUSYmbGVxO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZOLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0ZKRkZGLw==. Integration by parts is applied to the terms containing the derivatives, which then vanish as we can see from the following sketch. Integrals of the terms containing the derivatives can be written as 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM2LUkkSW50R0YlNiQqJi1JInVHRig2I0kieEdGKCIiIi1JJURpZmZHRiU2JComLCZGOUY5KiQpRjgiIiNGOSEiIkY5LUklZGlmZkdGJjYkLUkidkdGKEY3RjhGOUY4RjkvRjg7RkJGOTcjRjE= LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM3LUkkSW50R0YlNiQqJi1JInZHRig2I0kieEdGKCIiIi1JJURpZmZHRiU2JComLCZGOUY5KiQpRjgiIiNGOSEiIkY5LUklZGlmZkdGJjYkLUkidUdGKEY3RjhGOUY4RjkvRjg7RkJGOTcjRjE= Integration by parts and subtraction then lead to 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 IiIh What remains is 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 If the eigenvalues LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRInVGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRInZGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== are different, then 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which implies orthogonality of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Thus, 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJrRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRidGOQ==as we see for 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 via the matrix of evaluations below. 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSdSVEFCTEVHRig2JSIqLysteCItSSdNQVRSSVhHRig2IzcoNygiIiMiIiFGNkY2RjZGNjcoRjYjRjUiIiRGNkY2RjZGNjcoRjZGNiNGNSIiJkY2RjZGNjcoRjZGNkY2I0Y1IiIoRjZGNjcoRjZGNkY2RjYjRjUiIipGNjcoRjZGNkY2RjZGNiNGNSIjNkknTWF0cml4R0YlNyMtRkY2Iy9JJCVpZEdGKEYv From this matrix we also infer that 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, a result Maple cannot show in general, as we see from 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 LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiQ2JEkia0dGJ0kieEdGJyIiIyIiIi9GLzshIiJGMQ==
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Fourier-Legendre Series</Font></Text-field> An integrable function 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can be represented by the Fourier-Legendre series 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 The coefficients 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for the Fourier-Legendre series of the function 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKiQpLCYiIiJGMSokKUkieEdGKCIiI0YxISIiI0YxRjVGMTcjRi4= are 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiIhLCQqJiMiIiIiIiVGNEkjUGlHRiZGNEY0NyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiIiIiIhNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiIjLCQqJiMiIiYiI0siIiJJI1BpR0YmRjYhIiI3I0Yx LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiIkIiIhNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiIlLCQqJiMiIioiJGMjIiIiSSNQaUdGJkY2ISIiNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjIiImIiIhNyNGMQ== A partial sum of the Fourier-Legendre series itself is given by 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRGTGZHRigsKComIyIiIiIiJUYxSSNQaUdGJkYxRjEqKCMiIiYiI0tGMUYzRjEtSSpMZWdlbmRyZVBHRiU2JCIiI0kieEdGKEYxISIiKigjIiIqIiRjI0YxRjNGMS1GOTYkRjJGPEYxRj03I0Yu or better still, by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJEZMZkYnRjZGOUZARitGKw== LCgqJiMiJFgnIiVbPyIiIkkjUGlHJSpwcm90ZWN0ZWRHRidGJyooIyIkMCIiJUM1RidGKEYnKUkieEc2IiIiI0YnISIiKigjIiQ6JEYmRidGKEYnKUYvIiIlRidGMg== Figure 4 compares graphs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and the partial sum of its Fourier-Legendre series. 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 Figure 4 Graphs of 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 (in black) and a partial sum of its Fourier-Legendre series (in red)
An integrable function 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 be represented by the Fourier-Legendre series 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where 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 The function 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 has for its Fourier-Legendre coefficients the numbers 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiIhLCQqJiMiIiIiIiVGNEkjUGlHRiZGNEY0NyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiIiIiIhNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiIjLCQqJiMiIiYiI0siIiJJI1BpR0YmRjYhIiI3I0Yx LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiIkIiIhNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiIlLCQqJiMiIioiJGMjIiIiSSNQaUdGJkY2ISIiNyNGMQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYkdGKDYjIiImIiIhNyNGMQ== and for a partial sum of its Fourier-Legendre series, the polynomial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiYtRiw2JVEkRkxnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1GIzYlLUYsNiVRKXNpbXBsaWZ5RidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2Jy1JK211bmRlcm92ZXJHRiQ2Jy1GPTYtUSYmU3VtO0YnRkBGQkZFL0ZIRjhGSS9GTEY4L0ZORjhGT0Zobi9GVVEsMC4xNjY2NjY3ZW1GJy1GIzYlLUYsNiVRImtGJ0Y2RjktRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtSSNtbkdGJDYkUSIwRidGQC1GZnA2JFEiNUYnRkAvJSdhY2NlbnRHRkQvJSxhY2NlbnR1bmRlckdGREYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSQ1LjBGJy8lJmRlcHRoR0ZlcS8lKmxpbmVicmVha0dRJWF1dG9GJy1GIzYmLUklbXN1YkdGJDYlLUYsNiVRImJGJ0Y2RjktRiM2I0ZfcC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GaG5Gam4tRiM2JS1GLDYlUSpMZWdlbmRyZVBGJy9GN0ZERkBGZW4tRlxvNiQtRiM2JkZfcC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRmhuL0ZVUSwwLjMzMzMzMzNlbUYnLUYjNiUtRiw2JVEkY29zRidGY3NGQEZlbi1GXG82JC1GIzYjLUYsNiVRKSZ2YXJwaGk7RidGY3NGQEZARitGQEYrRitGK0ZARitGK0Yr LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRGTGdHRigsJCooIyIjOiIlWz8iIiJJI1BpR0YmRjMsKCIjViEiIiomIiM5RjMpLUkkY29zR0YlNiNJKSZ2YXJwaGk7R0YoIiIjRjNGMyomIiNARjMpRjsiIiVGM0YzRjNGNzcjRi4= Figure 5 compares graphs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and the partial sum of its Fourier-Legendre series. 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 NigtJSdDVVJWRVNHNiU3UzckJCIiISEiIiQiIiEhIiI3JCQiMWZEMkx6eFpvISM8JCIxOi0mUVRGQyVvISM8NyQkIjFOcHgqRypmIUciISM7JCIyKD5LbSQqPjV4NyEjPDckJCIwNiNleEdtXT4hIzokIjFgUCgpZmNKUT4hIzs3JCQiMlhULD0zb15pIyEjPCQiMSlROEpgPl5mIyEjOzckJCIwXEQwW25rSCQhIzokIjF5VSQpbyIpM1BLISM7NyQkIjEiPVphJnolKT1SISM7JCIyd0liRS41JD5RISM8NyQkIjJ3Yld2KW9HalghIzwkIjBFXSdbVmIxVyEjOjckJCIxJTMqKkhibShIXyEjOyQiMldDUEhKNlkqXCEjPDckJCIxUFJyNCkzVCplISM7JCIwSihITVNyZWIhIzo3JCQiMSNbKXl5a1l4bCEjOyQiMXRDSWk7TjhoISM7NyQkIjFzJ29jR28kenIhIzskIjFIPnZiO0t5bCEjOzckJCIxUTA7Z3IncCZ5ISM7JCIxJ2U1RGV5SjIoISM7NyQkIjFHTXQ/JFt0YCkhIzskIjFmI0clZTNTUHYhIzs3JCQiMSNwMzBrQEk+KiEjOyQiMSQ9d0VqJ3leeiEjOzckJCIwQyJcSGVWKXkqISM6JCIxKW9EJlxtXylIKSEjOzckJCIyR1spKSopM1cnXDUhIzskIjAsTWt6Z0NuKSEjOjckJCIxJ1tAWFNAJzQ2ISM6JCIxWTNfWUZJYiopISM7NyQkIjJWaFAzRypRejYhIzskIjFjLC8+P3RWIyohIzs3JCQiMlUzKjN0YzlUNyEjOyQiMG8kRyEqW2JoJSohIzo3JCQiMiR5MERCQSEqMzghIzskIjEqKW88eEksZicqISM7NyQkIjJQbUs/W0FNUCIhIzskIjElKnlRVCRcZSEpKiEjOzckJCIxYUBYLkV1UzkhIzokIjFZUi96I1xiIioqISM7NyQkIjJRdSMpKipbakRdIiEjOyQiMVVdMEkvdHcqKiEjOzckJCIyN208eW1YI3A6ISM7JCIxIjQnR3h6KSoqKioqISM7NyQkIjJuWi4qNCg0JlE7ISM7JCIxdkIwWkszeCoqISM7NyQkIjJgS2BARS8pKXAiISM7JCIxb0c9ZDs9PSoqISM7NyQkIjJ4UjEvLkNSdyIhIzskIjF6XCQ+USgzOSkqISM7NyQkIjB0JilINyo+Sj0hIzkkIjFWVHUnW2pHbSohIzs3JCQiMiRmYjd3WSwoKj0hIzskIjFtMEpOKjRFWiohIzs3JCQiMlkuaHpnJXBnPiEjOyQiMURmJVJGeCVcIyohIzs3JCQiMXBKODouU0o/ISM6JCIxYWZAWT4leSYqKSEjOzckJCIvMnpQRCRcNCMhIzgkIjElPidHNWNjZCcpISM7NyQkIjJ4LGZodW1GOyMhIzskIjFtbyI+LUclKUgpISM7NyQkIjJbWCczQFlCQ0EhIzskIjFLciE0KUcpKlJ6ISM7NyQkIjEvYjxXX1YiSCMhIzokIjEiKnkjUiRHJVFeKCEjOzckJCIwR05NemxZTiMhIzkkIjFKPyZRMXA9MyghIzs3JCQiMm1PbWskKmYyVSMhIzskIjBXKCl5JVs1K20hIzo3JCQiMTEoKT1VIXpgWyMhIzokIjFvWSpcWTc3NSchIzs3JCQiMWtISGQjSEliIyEjOiQiMTxXXG5ybV5iISM7NyQkIjFfciZbWCU9PUUhIzokIjJuITM5bCVcJCkqXCEjPDckJCIxSy0lXDA6W28jISM6JCIxIlJuQm0kZTVXISM7NyQkIjIwYzgrI1IqM3YjISM7JCIyL0ppRihlTTNRISM8NyQkIjEvMExNTmg2RyEjOiQiMnVdJGZVRkJTSyEjPDckJCIyR29VWDEwNylHISM7JCIxPmFWWjRidUQhIzs3JCQiMiU0NnhlJltNJUghIzskIjJsVkYweismbz4hIzw3JCQiMkw1IillNTgpNEkhIzskIjJMZ0I3byUpUkoiISM8NyQkIjJMeCFSWENMdEkhIzskIjF0X3dJNnM/byEjPDckJCItWF1FZlRKISM2JCIyWmwocE9NelJKISNELSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpUSElDS05FU1NHNiMiIiYtJSdDVVJWRVNHNiU3VzckJCIiISEiIiQiMiRwRllYcHhTPSEjPDckJCIyJXppYG0qKSlRVSQhIz0kIjFSIkghRysoZSY9ISM7NyQkIjFmRDJMenhabyEjPCQiMlZEJjRAIioqNCE+ISM8NyQkIjFiNFU6YSlvIykqISM8JCIyMUVFPSE0T2s+ISM8NyQkIjFOcHgqRypmIUciISM7JCIyS3QqKUdyXidcPyEjPDckJCIwNiNleEdtXT4hIzokIjEjUT88KjMyPUIhIzs3JCQiMlhULD0zb15pIyEjPCQiMTFubzYnXG1vIyEjOzckJCIwXEQwW25rSCQhIzokIjJhKnlXNjMmejgkISM8NyQkIjEiPVphJnolKT1SISM7JCIxRS54JEd1dmgkISM7NyQkIjJ3Yld2KW9HalghIzwkIjEmKXBcaVdrZ1QhIzs3JCQiMSUzKipIYm0oSF8hIzskIjFlR3BHVFxhWiEjOzckJCIxUFJyNCkzVCplISM7JCIxJUc2O1JTNE8mISM7NyQkIjEjWyl5eWtZeGwhIzskIjFRLScqW3VhIilmISM7NyQkIjFzJ29jR28kenIhIzskIjE3biVSYSMpPl4nISM7NyQkIjFRMDtncidwJnkhIzskIjFfSm87Lip5MighIzs3JCQiMUdNdD8kW3RgKSEjOyQiMSdII3kqNCNwLHchIzs3JCQiMSNwMzBrQEk+KiEjOyQiMUEkR2o+T2swKSEjOzckJCIwQyJcSGVWKXkqISM6JCIxTnZ1aXUjR1UpISM7NyQkIjJHWykpKikzVydcNSEjOyQiMTdSLUYlUigpeikhIzs3JCQiMSdbQFhTQCc0NiEjOiQiMTBkMzgweW0hKiEjOzckJCIyVmhQM0cqUXo2ISM7JCIxUWtmeVgsQiQqISM7NyQkIjJVMyozdGM5VDchIzskIjEkKj4iXExdTl0qISM7NyQkIjIkeTBEQkEhKjM4ISM7JCIxSiU9XSUpPmxsKiEjOzckJCIyUG1LP1tBTVAiISM7JCIxNitOYCdmSncqISM7NyQkIjFhQFguRXVTOSEjOiQiMW5yJVt1SCdRKSohIzs3JCQiMlF1IykqKltqRF0iISM7JCIxWnJhIT0oNHopKiEjOzckJCIyN208eW1YI3A6ISM7JCIxKio+JWZMb1QqKSohIzs3JCQiMm5aLio0KDQmUTshIzskIjFrd1Evckt6KSohIzs3JCQiMmBLYEBFLykpcCIhIzskIjA0Rm86JlJTKSohIzo3JCQiMnhSMS8uQ1J3IiEjOyQiMWQ0VVtWJipvKCohIzs3JCQiMHQmKUg3Kj5KPSEjOSQiMSoqNGlOZFJmJyohIzs3JCQiMiRmYjd3WSwoKj0hIzskIjFodSp6IT1SNyYqISM7NyQkIjJZLmh6ZyVwZz4hIzskIjEpb21DNEV6SyohIzs3JCQiMXBKODouU0o/ISM6JCIxTGlgOF82cCEqISM7NyQkIi8yelBEJFw0IyEjOCQiMUdlIik9ITNVeSkhIzs3JCQiMngsZmh1bUY7IyEjOyQiMGgmKnlaREZVKSEjOjckJCIyW1gnM0BZQkNBISM7JCIxRk9oVDRyViEpISM7NyQkIjEvYjxXX1YiSCMhIzokIjFPTncoPTVhZCghIzs3JCQiMEdOTXpsWU4jISM5JCIxLDYxeWl5KDMoISM7NyQkIjJtT21rJCpmMlUjISM7JCIxdmZUV3QjcGAnISM7NyQkIjExKCk9VSF6YFsjISM6JCIxOGRRI0cneW5mISM7NyQkIjFrSEhkI0hJYiMhIzokIjFeRicqR3A9YGAhIzs3JCQiMV9yJltYJT09RSEjOiQiMkYrWj5GMSVlWiEjPDckJCIxSy0lXDA6W28jISM6JCIyO0IrVilcYmtUISM8NyQkIjIwYzgrI1IqM3YjISM7JCIyYEFvdWx0emckISM8NyQkIjEvMExNTmg2RyEjOiQiMTopPVshKm8uOSQhIzs3JCQiMkdvVVgxMDcpRyEjOyQiMk8mR0ZkWmp0RSEjPDckJCIyJTQ2eGUmW00lSCEjOyQiMkVSOkNdSkdMIyEjPDckJCIyTDUiKWU1OCk0SSEjOyQiMjwxWnV6NT0xIyEjPDckJCIxUWZqdkZkVEkhIzokIjJFcmt6IUh4bz4hIzw3JCQiMkx4IVJYQ0x0SSEjOyQiMms5S3RJPTEhPiEjPDckJCIyalE/emFpdTUkISM7JCIyamdIPE51ZCY9ISM8NyQkIi1YXUVmVEohIzYkIjIkcEZZWHB4Uz0hIzwtJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkIiIhISIiJCIiISEiIi0lKlRISUNLTkVTU0c2IyIiIy0lJVZJRVdHNiQ7JCIiISEiIiQiK2FFZlRKISIqJShERUZBVUxURy0lKkFYRVNTVFlMRUc2IyUnTk9STUFMRy0lKFNDQUxJTkdHNiMlLENPTlNUUkFJTkVERy0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJDUkISIiLSUpQk9VTkRTX1lHNiMkIiVJOSEiIi0lLUJPVU5EU19XSURUSEc2IyQiJSFmJCEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiVTNiEiIi0lKUNISUxEUkVORzYi Figure 5 Graphs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTgxO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj8tSSNtb0dGJDYtUSI9RidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtRiw2JVEkc2luRidGPUY/RjVGPw== (in black) and a partial sum of its Fourier-Legendre series (in red)
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Azimuthal Asymmetry</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Separation of Variables</Font></Text-field> Without symmetry, 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so the separated form of the solution of Laplace's equation would be 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJVR0YoKigtSSRSaG9HRig2I0kkcmhvR0YoIiIiLUkkUGhpR0YoNiNJKSZ2YXJwaGk7R0YoRjMtSSZUaGV0YUdGKDYjSSZ0aGV0YUdGKEYzNyNGLg== Maple provides the following ODEs governing these three functions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocGRzb2x2ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRI0xFRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYn 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 We proceed to obtain these results from first principles. Upon division by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and multiplication by 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Laplace's equation becomes 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 If the terms containing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are moved to the right, we have 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 Introducing the separation constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy leads to the two equations 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM6LywqKiwiIiMiIiJJJHJob0dGKEY1KS1JJHNpbkdGJTYjSSkmdmFycGhpO0dGKEY0RjUtSSRSaG9HRig2I0Y2ISIiLUklZGlmZkdGJjYkRjxGNkY1RjUqKilGNkY0RjVGN0Y1RjxGPy1GQTYkRjwtSSIkR0YmNiRGNkY0RjVGNSoqRjhGNS1JJFBoaUdGKEY6Rj8tSSRjb3NHRiVGOkY1LUZBNiRGS0Y7RjVGNSooRjdGNUZLRj8tRkE2JEZLLUZINiRGO0Y0RjVGNUkjbXVHRig3Iy8sKkYzRjUqKkZERjVGN0Y1RjxGPy1GQTYkRkBGNkY1RjVGSkY1KihGN0Y1RktGPy1GQTYkRk9GO0Y1RjVGVg== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM7LywkKiYtSSZUaGV0YUdGKDYjSSZ0aGV0YUdGKCEiIi1JJWRpZmZHRiY2JEY0LUkiJEdGJjYkRjciIiMiIiJGOEkjbXVHRig3Iy8sJComRjRGOC1GOjYkLUY6NiRGNEY3RjdGQEY4RkE= The resulting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy-equation can be put into the form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM8LywmLUklZGlmZkdGJjYkLUkmVGhldGFHRig2I0kmdGhldGFHRigtSSIkR0YmNiRGOSIiIyIiIiomRjZGPkkjbXVHRihGPkY+IiIhNyMvLCYtRjQ2JC1GNDYkRjZGOUY5Rj5GP0Y+RkE= Continuity requires the imposition of the periodic boundary conditions 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 thus forming a Sturm-Liouville eigenvalue problem for which the solution is 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 Thus, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdXBHRiQ2JS1GLDYlUSJuRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjJGJ0YyRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=. Making this change in the companion equation, and dividing by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnRjVGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRi82JVEnJiM5NjY7RidGMkY1LUkjbW9HRiQ2LVEiLEYnRjUvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y1we have 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM9LywqKioiIiMiIiJJJHJob0dGKEY1LUkkUmhvR0YoNiNGNiEiIi1JJWRpZmZHRiY2JEY3RjZGNUY1KigpRjZGNEY1RjdGOi1GPDYkRjctSSIkR0YmNiRGNkY0RjVGNSoqLUkkUGhpR0YoNiNJKSZ2YXJwaGk7R0YoRjotSSRzaW5HRiVGSEY6LUkkY29zR0YlRkhGNS1GPDYkRkZGSUY1RjUqJkZGRjotRjw2JEZGLUZDNiRGSUY0RjVGNSomKUZKRjRGOilJIm5HRihGNEY1NyMvLCpGM0Y1KihGP0Y1RjdGOi1GPDYkRjtGNkY1RjVGRUY1KiZGRkY6LUY8NiRGTkZJRjVGNUZV Isolating the terms in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy yields the separated equation 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM+LywmKioiIiMiIiJJJHJob0dGKEY1LUkkUmhvR0YoNiNGNiEiIi1JJWRpZmZHRiY2JEY3RjZGNUY1KigpRjZGNEY1RjdGOi1GPDYkRjctSSIkR0YmNiRGNkY0RjVGNSwoKiYpLUkkc2luR0YlNiNJKSZ2YXJwaGk7R0YoRjRGOilJIm5HRihGNEY1RjUqKi1JJFBoaUdGKEZKRjpGSEY6LUkkY29zR0YlRkpGNS1GPDYkRk9GS0Y1RjoqJkZPRjotRjw2JEZPLUZDNiRGS0Y0RjVGOjcjLywmRjNGNSooRj9GNUY3RjotRjw2JEY7RjZGNUY1LChGRkY1Rk5GOiomRk9GOi1GPDYkRlNGS0Y1Rjo= and introduction of the separation constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy leads to the two ODEs 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiM/LywmKioiIiMiIiJJJHJob0dGKEY1LUkkUmhvR0YoNiNGNiEiIi1JJWRpZmZHRiY2JEY3RjZGNUY1KigpRjZGNEY1RjdGOi1GPDYkRjctSSIkR0YmNiRGNkY0RjVGNUknbGFtYmRhR0YoNyMvLCZGM0Y1KihGP0Y1RjdGOi1GPDYkRjtGNkY1RjVGRQ== LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiNALywoKiYpLUkkc2luR0YlNiNJKSZ2YXJwaGk7R0YoIiIjISIiKUkibkdGKEY5IiIiRj0qKi1JJFBoaUdGKEY3RjpGNUY6LUkkY29zR0YlRjdGPS1JJWRpZmZHRiY2JEY/RjhGPUY6KiZGP0Y6LUZENiRGPy1JIiRHRiY2JEY4RjlGPUY6SSdsYW1iZGFHRig3Iy8sKEYzRj1GPkY6KiZGP0Y6LUZENiRGQ0Y4Rj1GOkZM The LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy-equation can be manipulated to the form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiNBLywoKigiIiMiIiJJJHJob0dGKEY1LUklZGlmZkdGJjYkLUkkUmhvR0YoNiNGNkY2RjVGNSomKUY2RjRGNS1GODYkRjotSSIkR0YmNiRGNkY0RjVGNSomRjpGNUknbGFtYmRhR0YoRjUhIiIiIiE3Iy8sKEYzRjUqJkY+RjUtRjg2JEY3RjZGNUY1RkRGRkZH from which we see that it is an Euler equation solvable in powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy. The remaining ODE is the associated Legendre equation, which we cast in the form 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiNCLywqKigtSSRzaW5HRiU2I0kpJnZhcnBoaTtHRighIiItSSRQaGlHRihGNiIiIilJIm5HRigiIiNGO0Y7KiYtSSRjb3NHRiVGNkY7LUklZGlmZkdGJjYkRjlGN0Y7RjgqJkY0RjstRkM2JEY5LUkiJEdGJjYkRjdGPkY7RjgqKEY5RjtGNEY7SSdsYW1iZGFHRihGO0Y4IiIhNyMvLCpGM0Y7Rj9GOComRjRGOy1GQzYkRkJGN0Y7RjhGS0Y4Rk0= by bringing all terms to the left and multiplying through by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTY2O0YnRi9GMkYyRjItRiw2JVEkY29zRidGL0YyRjUtSSNtb0dGJDYtUSIuRidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlRGMg== The same change of variables that was used for Legendre's equation is applied, leading to 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiNDLywkKiYsMiomLUkieUdGKDYjSSJ4R0YoIiIiKUkibkdGKCIiI0Y6RjoqKEY9RjotSSVkaWZmR0YmNiRGNkY5RjpGOUY6RjoqKEY9RjopRjkiIiRGOkY/RjohIiItRkA2JEY2LUkiJEdGJjYkRjlGPUZFKihGPUY6RkZGOilGOUY9RjpGOiomRkZGOilGOSIiJUY6RkUqJkY2RjpJJ2xhbWJkYUdGKEY6RkUqKEY2RjpGUUY6RkxGOkY6RjopLCZGOkY6KiRGTEY6RkUjRjpGPUZFRkUiIiE3Iy8sJComLDJGNUY6Rj5GOkZCRkUtRkA2JEY/RjlGRSooRj1GOkZnbkY6RkxGOkY6KiZGZ25GOkZORjpGRUZQRkVGUkY6RjpGU0ZFRkVGVw== and then 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCgqJiwmIiIiRjAqJClJInhHRigiIiNGMCEiIkYwLUklZGlmZkdGJjYkLUkieUdGKDYjRjMtSSIkR0YmNiRGM0Y0RjBGMCooRjRGMC1GNzYkRjlGM0YwRjNGMEY1KigsKCokKUkibkdGKEY0RjBGMComSSdsYW1iZGFHRihGMEYyRjBGMEZIRjVGMCwmRjBGNUYxRjBGNUY5RjBGMCIiITcjLywoKiZGL0YwLUY3NiRGQEYzRjBGMEY/RjVGQkYwRko= after suitable rearrangement. A slightly better form for this equation can be obtained with the command 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicUdGKDYjIiNELywoKiYsJiIiIkY1KiQpSSJ4R0YoIiIjRjUhIiJGNS1JJWRpZmZHRiY2JC1JInlHRig2I0Y4LUkiJEdGJjYkRjhGOUY1RjUqKEY5RjUtRjw2JEY+RjhGNUY4RjVGOiomLCZJJ2xhbWJkYUdGKEY1KiYpSSJuR0YoRjlGNSwmRjVGOkY2RjVGOkY1RjVGPkY1RjUiIiE3Iy8sKComRjRGNS1GPDYkRkVGOEY1RjVGREY6RkdGNUZO but the form typically seen for Legendre's associated equation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNVEnaXRhbGljRictRiM2JC1GMTYkUSIyRidGNEY0LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y0RjQtRjg2LVEifkYnRjRGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZhby1GUzYlUSJ5RidGVkZZLUY4Ni1RIidGJ0Y0RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRidGYm9GZm8tRjg2LVEqJnVtaW51czA7RidGNEY7Rj5GQEZCRkRGRkZIRkpGTUZnbkZdb0ZSRl1vRmNvRmZvLUY4Ni1RIitGJ0Y0RjtGPkZARkJGREZGRkhGSkZNLUYsNiQtRiM2Ji1GUzYlUScmIzk1NTtGJy9GV0Y9RjRGNy1JJm1mcmFjR0YkNigtRlA2JS1GUzYlUSJuRidGVkZZRmVuRmpuRi4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmZxLyUpYmV2ZWxsZWRHRj1GNEY0Rl1vRmNvLUY4Ni1RIj1GJ0Y0RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GX3ItRjE2JFEiMEYnRjRGNA==
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">The Sturm-Liouville Eigenvalue Problem</Font></Text-field> The general solution of Legendre's associated equation is 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 Ly1JInlHNiI2I0kieEdGJSwmKiZJJF9DMUdGJSIiIi1JKkxlZ2VuZHJlUEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlLCYqJiNGKyIiI0YrKSwmKiYiIiVGK0knbGFtYmRhR0YlRitGK0YrRitGNEYrRitGNCEiIkkibkdGJUYnRitGKyomSSRfQzJHRiVGKy1JKkxlZ2VuZHJlUUdGLkYxRitGKw== a linear combination of the two associated Legendre functions, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRj1GP0ZBLUYsNiVRInhGJ0Y9Rj9GMkYyRjI= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRj1GP0ZBLUYsNiVRInhGJ0Y9Rj9GMkYyLUZCNi1RIi5GJ0YyRkUvRkhGMUZJRktGTUZPRlFGUy9GV0ZVRjI= (In extended typesetting mode, Maple writes these functions as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImFGJ0YyRjUvRjZRJ25vcm1hbEYnLUYjNiQtRi82JVEiYkYnRjJGNUY9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZGL0krbXNlbWFudGljc0dGJFEsW25vbmUsbm9uZV1GJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRInhGJ0YyRjVGPUY9LUkjbW9HRiQ2LVEiLEYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGWi8lKnN5bW1ldHJpY0dGWi8lKGxhcmdlb3BHRlovJS5tb3ZhYmxlbGltaXRzR0ZaLyUnYWNjZW50R0ZaLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGPQ==and 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respectively.) These are imaginatively called associated Legendre functions of the first and second kinds, respectively. From Legendre's associated equation we can see that 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEqTGVnZW5kcmVRRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRImFGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSIwRidGMkZBLUYsNiVRInhGJ0Y9Rj9GMkYyLUZCNi1RIj1GJ0YyRkUvRkhGMUZJRktGTUZPRlEvRlRRLDAuMjc3Nzc3OGVtRicvRldGX29GKy1GNjYkLUYjNiZGOkZBRmduRjJGMi1GQjYtUSIuRidGMkZFRl1vRklGS0ZNRk9GUUZTL0ZXRlVGMg== In the complex plane, a branch cut for a function is a line or line segment across which the function has a jump discontinuity. In Maple, there are two cut-regimes for the Legendre functions. The default regime imposes a cut on the real line coincident with the interval 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 Alternatively, the real intervals 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjg2LVEoJmluZmluO0YnRjRGOy9GP0Y9RkFGQ0ZFRkdGSUZLL0ZPRk1GNEY0RjQ= comprise a second cut regime. The environment variable _EnvLegendreCut is used to fix the cut regime by assigning it either of the expressions -1..1 or 1..infinity. To solve Laplace's equation in the interior of a sphere, the associated Legendre functions that arise must have their branch cut outside of the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIjFGJ0Y0LUYxNi1RIixGJ0Y0RjcvRjtRJXRydWVGJ0Y8Rj5GQEZCRkQvRkdRJjAuMGVtRicvRkpRLDAuMzMzMzMzM2VtRidGS0Y0RjRGT0Y0that is, opposite to the default regime. Hence, when working in Maple, the branch cut must be shifted via a proper assignment to the environment variable. However, commands such as evalf or simplify, commands that will most likely be invoked in the context of the solution process, have a remember table, which stores the value assigned to the environment variable. Reassigning a new value to the environment variable will not change the cut regime unless something is done to modify the remember tables in commands such as evalf and simplify. This is done by applying the forget command to these operators before changing the assignment to the environment variable. For the sake of completeness, we illustrate these issues below. With the default cut in place, the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUkjbW5HRiQ2JFEiNUYnRjItSSNtb0dGJDYtUSIsRidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjNGJ0YyRj4tRiw2JVEieEYnL0YwRkYvRjNRJ2l0YWxpY0YnRjJGMkYy is discontinuous across the line segment coincident with the real interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIjFGJ0Y0LUYxNi1RIixGJ0Y0RjcvRjtRJXRydWVGJ0Y8Rj5GQEZCRkQvRkdRJjAuMGVtRicvRkpRLDAuMzMzMzMzM2VtRidGS0Y0RjRGT0Y0a discontinuity we sample at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDAuNkYnRjlGOQ== with the evaluations 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 LCYkIisrK2c0NyEjOyIiIiomJCIrKys3QGchIilGJl4jRiZGJiEiIg== LCYkIisrK2c0NyEjOyIiIiomJCIrKys3QGchIilGJl4jRiZGJkYm Another way to see the discontinuity across this cut is symbolically, with 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 LCQqKiMiJDAiIiIjIiIiKSwmRichIiJJInhHNiJGJyMiIiRGJkYnKSwmRitGJ0YnRidGLUYnLCZGJ0YqKiYiIipGJylGK0YmRidGJ0YnRic= Careful inspection shows that for 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, the term 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 will be the square root of a negative number, from whence the discontinuity 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 LCQqJiwkXiMiIiJGJkYmKSwmRiZGJkkieEc2IiEiIiMiIiQiIiNGJkYr KiZeIyIiIkYkKSwmRiRGJEkieEc2IiEiIiMiIiQiIiNGJA== arises. Now, if we attempt to shift the branch cut with 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 it can appear that the behavior of simplify is erratic; simplify may or may not reflect the change in the branch cut, depending on the internal state of Maple. Here, we see 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 LCQqKiMiJDAiIiIjIiIiKSwmRichIiJJInhHNiJGJyMiIiRGJkYnKSwmRitGJ0YnRidGLUYnLCZGJ0YqKiYiIipGJylGK0YmRidGJ0YnRic= Thus, it is possible that we could have obtained exactly the same result as when the branch cut is along 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 The reason for the uncertainty lies in the remember table attached to the simplify command. Although reassignment to the environment variable is immediate, because of the remember table simplify will not immediately access the new value unless an internal event causes the table to be cleared. To force the remember table to access the new setting, use 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 the effect of which we test via 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 LCQqKiMiJDAiIiIjIiIiKSwmRidGJ0kieEc2IiEiIiMiIiRGJkYnKSwmRipGJ0YnRidGLUYnLCZGJ0YsKiYiIipGJylGKkYmRidGJ0YnRiw= In either event, notice that now the term 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 is real for 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and there will not be a jump across LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIjFGJ0Y0LUYxNi1RIixGJ0Y0RjcvRjtRJXRydWVGJ0Y8Rj5GQEZCRkQvRkdRJjAuMGVtRicvRkpRLDAuMzMzMzMzM2VtRidGS0Y0RjRGT0Y0as we see from 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 KiQpLCYiIiJGJUkieEc2IiEiIiMiIiQiIiNGJQ== KiQpLCYiIiJGJUkieEc2IiEiIiMiIiQiIiNGJQ==
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Relating Maple to the Literature</Font></Text-field> The associated Legendre function of the first kind appears in the literature with two different symbols. For example, in the Handbook of Mathematical Functions by Abramowitz and Stegun (Dover Publications), we find the following two formulas relating these functions to Legendre polynomials. 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 By the obvious experiment, we can conclude that in Maple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRIm5GJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEibUYnRj1GP0ZBLUYsNiVRInhGJ0Y9Rj9GMkYyLUZCNi1RIj1GJ0YyRkUvRkhGMUZJRktGTUZPRlEvRlRRLDAuMjc3Nzc3OGVtRicvRldGXm8tSShtc3Vic3VwR0YkNigtRiw2JVEiUEYnRj1GPy1GIzYkRjpGMi1GIzYkRllGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGXHAvSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUY2NiQtRiM2JEZmbkYyRjJGaW4tRiM2KC1JJW1zdXBHRiQ2JS1GNjYkLUYjNiUtRkI2LVEqJnVtaW51czA7RidGMkZFRlxvRklGS0ZNRk9GUS9GVFEsMC4yMjIyMjIyZW1GJy9GV0ZjcS1JI21uR0YkNiRRIjFGJ0YyRjJGMkZob0Zqby1GaXA2JS1GNjYkLUYjNiZGZXEtRkI2LVEoJm1pbnVzO0YnRjJGRUZcb0ZJRktGTUZPRlFGYnFGZHEtRmlwNiVGZm4tRiM2JC1GZnE2JFEiMkYnRjJGMkZqb0YyRjItRiM2JkZZLUZCNi1RIi9GJ0YyRkVGXG8vRkpGPkZLRk1GT0ZRL0ZUUSwwLjE2NjY2NjdlbUYnL0ZXRmBzRmZyRjJGam8tSSZtZnJhY0dGJDYoLUZpcDYlLUYsNiVRImRGJ0Y9Rj9GaG9Gam8tRiM2JC1GaXA2JS1GLDYlUSNkeEYnRj1GP0Zob0Zqb0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZmdC8lKWJldmVsbGVkR0YxLUklbXN1YkdGJDYlRmNvRmZvRl1wRmJwRjItRiw2I1EhRidGMg== Indeed, we construct LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJ0Y+LUYjNiQtRjs2JFEiM0YnRj5GPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGRy9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj5GPkY+ as 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 LCQqJiksJiIiIkYmKiQpSSJ4RzYiIiIjRiYhIiIjIiIkRitGJiwmKiYjIiRYKkYrRiZGKEYmRiYjIiQwIkYrRixGJkYs and compare it to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEqTGVnZW5kcmVQRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUkjbW5HRiQ2JFEiNUYnRjItSSNtb0dGJDYtUSIsRidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjNGJ0YyRj4tRiw2JVEieEYnL0YwRkYvRjNRJ2l0YWxpY0YnRjJGMkYy in the form 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 LCQqKiMiJDAiIiIjIiIiKSwmRidGJ0kieEc2IiEiIiMiIiRGJkYnKSwmRipGJ0YnRidGLUYnLCZGJ0YsKiYiIipGJylGKkYmRidGJ0YnRiw= On LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSIxRidGLy1GLDYtUSYmbGVxO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZOLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0ZKRkZGLw==, the radicals appearing in both expressions are equivalent, as demonstrated by the graphs in Figure 6. 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 Figure 6 Graphs of 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 (red) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJ0Y+LUYjNiQtRjs2JFEiM0YnRj5GPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGRy9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj5GPkY+ (black)
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Orthogonality of the Eigenfunctions</Font></Text-field> The orthogonality relation for the associated Legendre functions of the first kind is 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 a relation Maple can instantiate, but not easily establish from first principles. For example, Table 2 lists some integrals for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJuRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRidGOQ==while Table 3 lists some for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJuRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRidGOQ==comparing the computed and formulaic values of the integral. 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NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJSIiI0kieEdGKUYxIiIiL0YyOyEiIkYzIiMhKUY3 NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJSIiJEkieEdGKSIiIyIiIi9GMjshIiJGNCIlPzZGOA== NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJUYwSSJ4R0YpIiIjIiIiL0YxOyEiIkYzIiVnKilGNw== NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJiIiIkkieEdGKSIiI0YxL0YyOyEiIkYxIyIjZyIjNkY3 NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJiIiI0kieEdGKUYxIiIiL0YyOyEiIkYzIyIlIW8iIiM2Rjc= NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJiIiJEkieEdGKSIiIyIiIi9GMjshIiJGNCMiJj8uJSIjNkY4 NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJiIiJUkieEdGKSIiIyIiIi9GMjshIiJGNCMiJ2dkcyIjNkY4 NiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSpMZWdlbmRyZVBHRiY2JSIiJkYwSSJ4R0YpIiIjIiIiL0YxOyEiIkYzIyIoK3dEKCIjNkY3 Table 2 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiUtRi82LVEqJnVtaW51czA7RidGMi9GNlEmZmFsc2VGJy9GOUZQL0Y7RlAvRj5GUC9GQEZQL0ZCRlAvRkRGUC9GRlEsMC4yMjIyMjIyZW1GJy9GSUZYLUkjbW5HRiQ2JFEiMUYnRjJGMi1GIzYkRlpGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGXG8tRiw2KC1JI21pR0YkNiVRIlBGJy8lJ2l0YWxpY0dGPC9GM1EnaXRhbGljRictRiM2JC1GYm82JVEibkYnRmVvRmdvRjItRiM2JC1GYm82JVEibUYnRmVvRmdvRjJGam5GXW8vSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRmJvNiVRInhGJ0Zlb0Znb0YyRjItRi82LVEifkYnRjJGT0ZRRlJGU0ZURlVGVkZFRkgtRiw2KEZhby1GIzYkLUZibzYlUSJrRidGZW9GZ29GMkZecEZqbkZdb0ZjcEZecS1GLzYuUTAmRGlmZmVyZW50aWFsRDtGJ0Zlb0Znb0Y1RjgvRjtGN0Y9L0ZARjdGQUZDRkVGSEZbcS1GLzYtUSI9RidGMkZPRlFGUkZTRlRGVUZWL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRmFyLUZlbjYkUSIwRidGMkYy for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJuRidGL0YyRjk= Table 3 For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJuRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRidGOQ==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 to 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
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Fourier-Legendre Series</Font></Text-field> The functions 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 with the functions 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form a complete set on the rectangle 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 Consequently, a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTY2O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RidGPUY/Rj9GP0Y/ can be expanded in a Fourier-Legendre series of the form 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= and 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For example, take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTY2O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RidGPUY/Rj9GP0Y/ as 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoLCQqLCMiIiIiIidGMUkpJnZhcnBoaTtHRihGMSwmSSNQaUdGJkYxRjMhIiJGMUkmdGhldGFHRihGMSwmKiYiIiNGMUY1RjFGMUY3RjZGMUYxNyNGLg== a function whose graph is seen in Figure 7. 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Figure 7 Graph of 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
It is also useful to define the function 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 Ly1JImRHNiI2JEkia0dGJUkibUdGJSwkKioiIiMiIiItSSpmYWN0b3JpYWxHJSpwcm90ZWN0ZWRHNiMsJkYnRixGKEYsRiwtRi42IywmRidGLEYoISIiRjUsJiomRitGLEYnRixGLEYsRixGNUYs whose values appear in the denominators of the expressions for the series coefficients. The computation of these coefficients is slightly simplified by recognizing that all the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI2ttRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= are zero by symmetry. Then, the first few LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRI2swRidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= are given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZHLUkjbW9HRiQ2L1EkZm9yRicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkgtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNWVtRicvJSZkZXB0aEdGUC8lKmxpbmVicmVha0dRJWF1dG9GJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dGMS9GM1EnaXRhbGljRidGSy1GLDYvUSVmcm9tRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGSy1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRidGSy1GLDYvUSN0b0YnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRkstRl9vNiRRIzEwRidGYm9GSy1GLDYvUSNkb0YnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJLUZMNiZGTkZRRlQvRldRNmluY3JlYXNlaW5kZW50bmV3bGluZUYnLUZMNiZGTi9GUlEmMC4wZW1GJ0ZUL0ZXUShuZXdsaW5lRictRkw2JkZORmNwRlRGVi1JJW1zdWJHRiQ2JS1GWjYlUSJBRidGZ25GaW4tRiM2JUZZLUYsNi1RIixGJ0Zib0Y3L0Y7RjFGPEY+RkBGQkZERkYvRkpRLDAuMzMzMzMzM2VtRidGXm8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYtUSomY29sb25lcTtGJ0Zib0Y3RjpGPEY+RkBGQkZEL0ZHUSwwLjI3Nzc3NzhlbUYnL0ZKRl5yLUkmbWZyYWNHRiQ2KC1GX282JFEiMUYnRmJvLUYjNigtRl9vNiRRIjJGJ0Ziby1GLDYtUSJ+RidGYm9GN0Y6RjxGPkZARkJGREZGRkktRlo2JVElJnBpO0YnL0ZobkY5RmJvRltzLUZaNiVRImRGJ0ZnbkZpbi1JKG1mZW5jZWRHRiQ2JEZfcUZiby8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXQvJSliZXZlbGxlZEdGOS1JKG1zdWJzdXBHRiQ2Jy1GLDYtUSYmaW50O0YnRmJvRjdGOkY8Rj5GQEZCRkRGRkZJRl5vLUYjNiVGaHItRiw2LVExJkludmlzaWJsZVRpbWVzO0YnRmJvRjdGOkY8Rj5GQEZCRkRGRkZJRl5zLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJ0ZncS1GY3Q2J0ZldEZeb0Zec0ZddUZncS1GWjYjUSFGJy1GIzYoLUZaNiVRImdGJ0ZnbkZpbkZqdC1GIzYlLUZaNiVRKXNpbXBsaWZ5RidGZ25GaW4tRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGYm9GN0Y6RjxGPkZARkJGREZGRkktRmZzNiQtRiM2JUZidS1GIzYlLUZaNiVRKkxlZ2VuZHJlUEYnRmFzRmJvRl92LUZmczYkLUYjNihGWUZhcUZeb0ZhcS1GIzYlLUZaNiVRJGNvc0YnRmFzRmJvRl92LUZmczYkLUYjNiMtRlo2JVEpJnZhcnBoaTtGJ0Zhc0Zib0Zib0ZidUZib0ZidUZib0ZqdC1GIzYlLUZaNiVRJHNpbkYnRmFzRmJvRl92RmR3RmJ1RmJ1LUZMNiZGTi9GUlEmMC4zZW1GJ0ZURlYtRiw2LlEwJkRpZmZlcmVudGlhbEQ7RidGZ25GaW5GN0Y6RjxGPkZARkJGREZGRklGaHdGYHhGZHgtRlo2JVEoJnRoZXRhO0YnRmFzRmJvLUZMNiZGTkZRRlQvRldRNmRlY3JlYXNlaW5kZW50bmV3bGluZUYnRmFwRltzLUYsNi9RJGVuZEYnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRltzRmpvLUYsNi1RIjpGJ0Zib0Y3RjpGPEY+RkBGQkZERl1yRl9y and the first few 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 are given by 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 A partial sum of the Fourier-Legendre series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is then 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 the graph of which can be seen in Figure 8. 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Figure 8 Graph of partial sum of the Fourier-Legendre series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRJyYjOTgxO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEnJiM5NTI7RidGPUY/Rj9GP0Y/
To estimate the accuracy of this approximation, the difference LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJnRidGL0YyRjk= is plotted in Figure 9. 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Figure 9 Graph of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJnRidGL0YyRjk= as an estimate of the accuracy of the partial sum LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=
Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 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