<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

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Classroom Tips and Techniques:

Roles for the Laplace Transform's Shifting Laws

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

JSFH

<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

JSFH

Table 1 lists three shifting laws for the Laplace transform, itself defined by the integral

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 JSFH

First Shifting Law

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

Second Shifting Law

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

Third Shifting Law

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

Table 1 Three shifting laws for the Laplace transform; in each, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8

The first shifting law might be loosely articulated as "multiplication by an exponential in the time domain causes a shift in the transform domain," while the second appears to be its converse: "multiplication by an exponential in the transform domain causes a shift in the time domain." This wonderful symmetry between the first and second shifting laws may be the reason why these two are more prominent in the classroom than the third. In fact, in all my years teaching Laplace transforms at the Rose-Hulman Institute of Technology, the third shifting law was never mandated as part of the syllabus. A statement of this law usually appeared on the Laplace transform "cheat sheet" students were allowed to use on departmental final exams, but it was left to individual instructors to cover it or not.

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For students unaware of the third shifting law, the Laplace transform of a function multiplied by 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 could be a significant challenge. They have to obtain the transform with the second shifting law, a process that typically requires algebraic virtuosity in whose absence the functioning of the law becomes opaque. It would be so much better to teach students to use the third shifting law for this application, and to use the second shifting law for inverting transforms that contain the "second shift flag," namely, the factor 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.

In this article we explore examples where the "forward" transform of 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 is obtained with the second shifting law, and compare the computational effort to that expended by doing the same transform with the third shifting law. Our conclusion is that students should learn the third shifting law for the "forward" transform, and use the second shifting law only for the inversion of transforms containing an exponential factor.

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<Text-field style="Heading 1" layout="Heading 1">Symmetry in the Operational Laws</Text-field>

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The four basic operational laws for Laplace transforms are listed in Table 2.

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

Table 2 Four operational laws for the Laplace transform

To within a detail for each law, each law can be articulated succinctly and symmetrically, as seen in Table 3.

Time Domain

Transform Domain

Differentiate with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Multiply by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Multiply by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Differentiate with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Integrate

Divide by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Divide by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Integrate with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn

Table 3 Symmetry in the four operational laws

I was fortunate to have been taught the Laplace transform this way in 1965, and I taught it the same way ever since. However, the "devil is in the details," and each law requires attention to an important detail. Thus, when transforming a derivative, you have additional terms, descending in powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==, and ascending in orders of differentiation. When transforming the product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ5RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiVGKy8lK2V4ZWN1dGFibGVHRj1GOUY5RldGOQ==, the differentiation in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== requires a minus sign. When transforming an integral, or a division by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==, the limits of integration in each case are significant.

Just as for the four basic operational laws, the first and second shifting laws exhibit an interesting symmetry. In the first law, transforming a product containing an exponential factor results in a shift in the transform domain, whereas in the second law, transforming a shift in the time domain results in a product with an exponential factor in the transform domain. But as for the operational laws, the devil is again in the details.

For the first shifting law, the detail would be the sign of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== in the factor 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relative to the direction of the shift in the transform. For the second shifting law, the detail is the Heaviside factor. Not only must the function 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 appear with shifted argument, it must also be multiplied by a Heaviside function with the same shifted argument. Applying the second shifting law in the "forward" direction, that is, using it to compute a transform, generally presents great difficulties for many students.

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<Text-field style="Heading 1" layout="Heading 1"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEoJiM4NDY2O0YnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNi0tRiw2KFEiZ0YnRi8vJSVib2xkR0Y0RjIvRjZRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUY5NiQtRiM2KS1GLDYoUSJ0RidGL0ZARjJGQkZEL0YwUSMxOEYnRkAvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKkhlYWRpbmd+MUYnL0Y2RkZGRC9GNlEnbm9ybWFsRictSSNtb0dGJDYtUSFGJ0ZXLyUmZmVuY2VHRlIvJSpzZXBhcmF0b3JHRlIvJSlzdHJldGNoeUdGUi8lKnN5bW1ldHJpY0dGUi8lKGxhcmdlb3BHRlIvJS5tb3ZhYmxlbGltaXRzR0ZSLyUnYWNjZW50R0ZSLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZ28tRlo2MFEqSGVhdmlzaWRlRidGL0ZARlZGREZnbkZpbkZbb0Zdb0Zfb0Zhb0Zjb0Zlb0Zoby1GOTYkLUYjNixGSy1GWjYwUSJ+RidGL0ZARlZGREZnbkZpbkZbb0Zdb0Zfb0Zhb0Zjb0Zlb0Zoby1GWjYwUSgmbWludXM7RidGL0ZARlZGREZnbkZpbkZbb0Zdb0Zfb0Zhb0Zjby9GZm9RLDAuMjIyMjIyMmVtRicvRmlvRmhwLUYsNihRImFGJ0YvRkBGMkZCRkRGTkZARlBGU0ZWRkRGV0ZORkBGUEZTRlZGREZXLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLyUnZmFtaWx5R1EmU2VyaWZGJ0ZORkBGUEZTRlZGRA==">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</Equation></Text-field>

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Computing the Laplace transform of the product 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 via the second shifting law requires algebraic virtuosity that must be acquired long before the student enters the differential equations course where the Laplace transform is encountered. Indeed, this virtuosity only works for a special class of functions, namely, polynomials, the exponential function, the sine and cosine functions, and for the really gifted, the hyperbolic sine and cosine functions.

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The second shifting law requires that 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 be manipulated into the form 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 by writing, and simplifying, 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 as 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 These transformations are even more difficult to implement in Maple since Maple automatically simplifies LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyLUY2Ni1RJyZwbHVzO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GTy8lK2V4ZWN1dGFibGVHRj1GOQ== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==. In Maple, such manipulations are often best expressed as text.

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<Text-field style="Heading 2" layout="Heading 2">Example 1</Text-field>

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The Laplace transform of 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 is easily obtained in Maple by accessing the laplace command through the Context Menu system. (Select Integral Transforms, then Laplace Transform.)

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

= KiYsKiomIiIkIiIiKUkidEc2IkYlRiZGJiomIiIjRiYpRihGK0YmRiYqJiIiJUYmRihGJkYmRiYhIiJGJi1JKkhlYXZpc2lkZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjLCZGKEYmRitGL0Ym

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMkxhcGxhY2V+dHJhbnNmb3JtRidGL0YyRjVGRQ==

inttrans[:-laplace](,t,s);

KigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiMiIiJJInNHRihGLSEiIkYtLCoqJiIjUkYtKUYuIiIkRi1GLSomIiNbRi0pRi5GLEYtRi0qJiIjU0YtRi5GLUYtIiM9Ri1GLSlGLiIiJUYv

To obtain this result by the second shifting law, each appearance of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in the cubic polynomial must be changed to 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 so that this polynomial becomes

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and then, after great pains, 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. The second shifting law then gives the results in Table 4

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

Table 4 Application of second shifting law

Of course, the sum of terms on the right in Table 4 agrees with the transform obtained with Maple's Laplace transform.

JSFH

To obtain this transform by the third shifting law, simply define the cubic polynomial as the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA= and multiply the transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiK0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1JI21uR0YkNiVRIjJGJ0YyRkRGMkZERjJGREYyRkQ= by the appropriate exponential factor. See Table 5.

Context Menu: Assign Function

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

= Ly1JImdHNiI2I0kidEdGJSwqKiYiIiQiIiIpRidGKkYrRisqJiIiI0YrKUYnRi5GK0YrKiYiIiVGK0YnRitGK0YrISIi

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJnRzYi

Context Menu: Integral Transforms\342\211\273Laplace Transform

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInRGJ0YvRjItSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIyRidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBRmVuRkE=

= LCoqJiIiJCIiIiksJkkidEc2IkYlIiIjRiVGJEYlRiUqJkYqRiUpRidGKkYlRiUqJiIiJUYlRihGJUYlIiIoRiU=

inttrans[:-laplace](,t,s);

KiYsKiomIiNSIiIiKUkic0c2IiIiJEYmRiYqJiIjW0YmKUYoIiIjRiZGJiomIiNTRiZGKEYmRiYiIz1GJkYmKUYoIiIlISIi

Table 5 Application of the third shifting law

To complete the calculation, simply multiply the Laplace transform in Table 5 by the exponential factor 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.

It would then appear that using the second shifting law to obtain the Laplace transform of a function in the form 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 requires a great deal of algebraic virtuosity. It is so much easier to use the third shifting law, and to reserve the second shifting law for the inversion of transforms whose form is 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.

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<Text-field style="Heading 2" layout="Heading 2">Example 2</Text-field>

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The Laplace transform of 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 is easily obtained in Maple by accessing the laplace command through the Context Menu system. (Select Integral Transforms, then Laplace Transform.)

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

= KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiQiIiJJInRHRihGLSEiIkYtLUkqSGVhdmlzaWRlR0YlNiMsJkYuRi0iIiNGL0Yt

inttrans[:-laplace](,t,s);

KiYsJkkic0c2IiIiIiIiJEYmISIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiIiJ0YoKiYiIiNGJkYkRiZGKEYm

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To obtain this result by the second shifting law, each appearance of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== in the exponential term must be changed to 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 so that this factor becomes

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The second shifting law then gives

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

Table 6 Application of the second shifting law

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The result in Table 6 matches the result given by Maple's Laplace transform.

To obtain this transform by the third shifting law, simply define the exponential factor as the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= and multiply the transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInRGJ0YvRjItSSNtb0dGJDYtUSIrRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIyRidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBRmVuRkE= by 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. See Table 7.

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Context Menu: Unassign

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

= SSJnRzYi

unassign(''); ;

SSJnRzYi

Context Menu: Assign Function

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

= Ly1JImdHNiI2I0kidEdGJS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJiIiJCIiIkYnRjEhIiI=

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

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Context Menu: Integral Transforms\342\211\273Laplace Transform

= LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiomIiIkIiIiSSJ0R0YnRiwhIiIiIidGLg==

inttrans[:-laplace](,t,s);

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyEiJyIiIiwmSSJzR0YoRisiIiRGKyEiIg==

Table 7 Application of the third shifting law

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To complete the calculation, simply multiply the Laplace transform in Table 7 by the exponential factor 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.

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<Text-field style="Heading 2" layout="Heading 2">Example 3</Text-field>

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The Laplace transform of 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 is easily obtained in Maple by accessing the laplace command through the Context Menu system. (Select Integral Transforms, then Laplace Transform.)

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

= KiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiYiIiQiIiJJInRHRihGLUYtRi0tSSpIZWF2aXNpZGVHRiU2IywmRi5GLSIiIyEiIkYt

inttrans[:-laplace](,t,s);

LCQqKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiIiIyIiIkkic0dGKUYuISIiRi4sJiomLUkkc2luR0YmNiMiIidGLkYvRi5GMComIiIkRi4tSSRjb3NHRiZGNUYuRjBGLiwmKiQpRi9GLUYuRi4iIipGLkYwRjA=

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To obtain this result by the second shifting law, each appearance of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== in the sine term must be changed to 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 so that this trig function becomes

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The second shifting law then gives the results in Table 8.

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

Table 8 Application of the second shifting law

Of course, the sum of terms on the right in Table 8 agrees with the transform obtained with Maple's Laplace transform.

To obtain this transform by the third shifting law, simply define the function 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 and multiply the transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEidEYnRi9GMkY1LUkjbW9HRiQ2LlEiK0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1JI21uR0YkNiVRIjJGJ0YyRkRGMkZERjJGREYyRkQ= by the appropriate exponential factor. See Table 9.

JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkM=

Context Menu: Unassign

= SSJnRzYi

unassign(''); ;

SSJnRzYi

Context Menu: Assign Function

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

= Ly1JImdHNiI2I0kidEdGJS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJiIiJCIiIkYnRjFGMQ==

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJnRzYi

Context Menu: Integral Transforms\342\211\273Laplace Transform

= LUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJiomIiIkIiIiSSJ0R0YnRixGLCIiJ0Ys

inttrans[:-laplace](,t,s);

LCQqJiwmKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyIiJyIiIkkic0dGK0YuISIiKiYiIiRGLi1JJGNvc0dGKEYsRi5GMEYuLCYqJClGLyIiI0YuRi4iIipGLkYwRjA=

Table 9 Application of the third shifting law

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkM=

To complete the calculation, simply multiply the Laplace transform in Table 9 by the exponential factor 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.

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<Text-field style="Heading 1" layout="Heading 1"><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation></Text-field>

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Inverting the Laplace transform 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 by the second shifting law is far simpler than using this law for computing a transform in the "forward" direction. Once the factor 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 is recognized as the "second shift" flag, that is, as the signal that the second shifting law must be used to obtain the inverse transform, the inversion steps can be articulated by the algorithm given in Table 10.

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1. Ignore the factor 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 and invert 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 to 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

2. Shift 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 to 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

3. Multiply 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 by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2KC1GLDYuUSpIZWF2aXNpZGVGJy8lK2V4ZWN1dGFibGVHRjRGL0YyL0Y2USV0cnVlRidGN0Y5RjtGPUY/RkEvRkVRLDAuMzMzMzMzM2VtRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1JKG1mZW5jZWRHRiQ2JS1GIzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR0ZORksvRjBRJ2l0YWxpY0YnLUYsNi5RKCZtaW51cztGJ0ZLRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZfby1GWjYmUSJhRidGZ25GS0ZpbkZLRi9GS0YvRitGS0YvRitGS0Yv

Table 10 Inversion by second shifting law

Implementing the steps in Table 10 requires no special algebraic skill, no recollection of function-specific properties. The inversion process is as algorithmic with the second shifting law as the transform process is with the third shifting law.

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<Text-field style="Heading 2" layout="Heading 2">Example 4</Text-field>

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The inverse of the Laplace transform 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 is easily obtained in Maple by accessing the invlaplace command through the Context Menu system. (Select Integral Transforms, then Inverse Laplace Transform.)

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2Jy1JJm1mcmFjR0YkNigtRiM2Jy1JI21pR0YkNiZRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjBRJ2l0YWxpY0YnLUYsNi5RIitGJ0ZURi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZnbi1JI21uR0YkNiVRIjFGJ0ZURi9GUUZXLUYjNictSSVtc3VwR0YkNiVGTS1GIzYlLUZqbjYlUSIyRidGVEYvRlFGVy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGWUZpbkZRRlcvJS5saW5ldGhpY2tuZXNzR0Zcby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZecC8lKWJldmVsbGVkR0Y0LUYsNi5RIn5GJ0ZURi9GMkY1RjdGOUY7Rj1GP0ZBRkQtRmBvNiUtRiM2KEYrLUYjNiVGKy1GLDYtUS8mRXhwb25lbnRpYWxFO0YnRi8vRjNRJnVuc2V0RicvRjZGYHEvRjhGYHEvRjpGYHEvRjxGYHEvRj5GYHEvRkBGYHFGQUZERi9GK0ZRRlRGVy1GIzYoLUYsNi5RKiZ1bWludXMwO0YnRlRGL0YyRjVGN0Y5RjtGPUY/RmZuRmhuLUZqbjYlUSIzRidGVEYvRmNwRk1GUUZXRmdvLyUrZXhlY3V0YWJsZUdGNEYvRitGX3JGLw==

= KigsJkkic0c2IiIiIkYmRiZGJiwmKiQpRiQiIiNGJkYmRiZGJiEiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJiIiJEYmRiRGJkYrRiY=

inttrans[:-invlaplace](,s,t);

KiYtSSpIZWF2aXNpZGVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJ0R0YoIiIiIiIkISIiRiwsJi1JJGNvc0dGJUYpRiwtSSRzaW5HRiVGKUYsRiw=

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To obtain this result via first principles, write the rational function as

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and recognize the first fraction as the transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA==; the second, as the transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA==. Sum these terms, replace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEoJm1pbnVzO0YnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjNGJ0YyRjxGMkY8 and multiply the resulting expression by 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. This will result in exactly the expression provided by Maple.

So in summary, use the second shifting law to invert transforms containing an exponential factor, but use the third shifting law to transform functions containing a Heaviside factor.

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