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26 - Lecture 26 We now return to special functions and...

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Lecture 26 We now return to special functions and complete our ‘trinity’ with an elementary account of Bessel functions including one of their connexions with Lie algebra representation theory. We begin with an origin of Bessel functions (one among many): they arise in the search for eigenfunctions of the Laplacian in polar coordinates by separation of variables; these may be cylindrical or spherical coordinates for the spatial Laplacian, but we shall simply consider the planar Laplacian Δ = 2 r + 1 r r + 1 r 2 2 θ . By direct substitution, the function f given in polar coordinates by f ( r, θ ) = R ( r )Θ( θ ) satisfies Δ f = - λf if and only if R 00 ( r )Θ( θ ) + 1 r R 0 ( r )Θ( θ ) + 1 r 2 R ( r 00 ( θ ) = - λR ( r )Θ( θ ) . Division by R ( r )Θ( θ ) and multiplication by r 2 lead to the equation r 2 R 00 ( r ) R ( r ) + r R 0 ( r ) R ( r ) + λr 2 = - Θ 00 ( θ ) Θ( θ ) . Here, the LHS depends on r only while the RHS depends on θ only; ac- cordingly, both sides have a constant value b . In order that the resulting Θ equation Θ 00 = - b Θ 1
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