Separation of Variables
Under the assumption that the steadystate temperatures are symmetric about the axis, dependence on angle can be dispensed with. Hence, and a Maplegenerated variableseparation is obtained with
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(6.1.1) 
Equation <$/I%diffGF$6$F96$F..." align="center" border="0"> shows that a variable separation solution of the form
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exists, and provides the ordinary differential equations the functions and 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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Moving all terms in to the right, we then have
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Introduction of Bernoulli's separation constant then leads to the ordinary differential equations
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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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and then
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Now, make the change of variables with becoming This is done in Maple with
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Further simplifying, we have
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which is the standard form of Legendre's equation, the selfadjoint form of which would be
The SturmLiouville Eigenvalue Problem
The eigenvalue problem that embeds Legendre's equation is singular. The boundary conditions are simply that must be continuous on the interval Passage from the general solution
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to the eigenfunctions is surprisingly more difficult than it was for Bessel's equation. Because we are in extended typesetting mode, the functions and are displayed as and respectively. (Were we in extended typesetting mode during our earlier discussion of Bessel's equation, Maple would have displayed as )
When solving Laplace's equation in the cylinder, it was relatively easy to use continuity to restrict the general solution to just , the Bessel function bounded on the interval and to determine the eigenvalues from the zeros of 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 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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from which it is clear that the function is unbounded at the endpoints because of the logarithms. But this is obvious for , an integer. It is a bit more difficult to divine the endpoint behavior for general values of . For example, we can calculate the values
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which suggest may indeed be unbounded at for general values of . Figure 1 contains graphs of the real and imaginary parts of with in the open interval and in the interval .
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Figure 1 Real and imaginary parts of for suggesting is unbounded on

From Figure 1(a) especially, we conclude that is unbounded for general values of . On the basis of this conclusion, we set to zero in the general solution of Legendre's equation, and turn our attention to the Legendre function of the first kind.
We first show that for general (real) values of is unbounded. Sample calculations include
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To illustrate this behavior for multiple values of , we define the following piecewise function.
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If is large, then a graph of will show a point at for that value of . If is "not large" then a graph of will show the value of
We can control the evaluation points for a graph of if we define the uniform random variable via
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then create a uniform but random sample of values that includes the integers in the interval .
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The graph of in Figure 2 shows that virtually all evaluations of are large in magnitude.
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Figure 2 Stylized graph of for

However, it also suggests that for integer . For noninteger , is unbounded so that the bounded solutions of Legendre's equation
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will be the eigenfunctions with
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(6.2.1) 
an integer. Hence, the eigenvalues will be
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that is, The first few eigenfunctions are
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which are the Legendre polynomials normalized so that These polynomials are graphed in Figure 3.
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Figure 3 The Legendre polynomials

That the function reduces to the polynomial for can be seen from the following calculations.
For noninteger , we first obtain the formal power series expansion of via
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then extract the general term in the first series with
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The pochhammer symbol
or "rising factorial" for complex generalizes to
for complex If is a nonpositive integer, then
Making this transformation and setting in the general term of the first series for gives the general coefficient
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For large this coefficient is asymptotic to
suggesting that is unbounded since the series under consideration will behave like the harmonic series at . We can confirm this behavior by comparing the general coefficient with for large . In the limit we find the ratio tends to
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which is finite for not an integer. To see that for integer the series for reduces to a polynomial, examine the recursion formula for its coefficients. This is most efficiently obtained in Maple via
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from which it becomes clear that when Hence, Therefore, is a polynomial of degree for .
Orthogonality of the Eigenfunctions
The classical proof of the orthogonality of the eigenfunctions of Legendre's equation is based on integration by parts. The selfadjoint form of the equation, namely,
is written once for an eigenfunction and once for The first equation is multiplied by and the second, by , and the difference of the two products is integrated over . 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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Integration by parts and subtraction then lead to
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What remains is If the eigenvalues and are different, then which implies orthogonality of and .
Thus, for as we see for via the matrix of evaluations below.
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From this matrix we also infer that , a result Maple cannot show in general, as we see from
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