# Bessel functions

Problem:

(1) Show that there exist exist polynomials P_n(z) and Q_n(z) for positive integers n such that the Bessel function J_{n + 1/2}(x) satisfies

J_{n + 1/2}(x) = P_n(x^{-1/2}) sin x + Q_n(x^{-1/2}) cos x.

(2) Find ( P_1(z) ) and ( Q_1(z) ).

This problem tells us that Bessel functions of half-integer order can be expressed in terms of polynomials and the sin and cos functions. The main idea of the solution is to use a recursion relation and the Bessel functions J_{1/2} and J_{-1/2} in conjunction with a proof by induction.

For some background and more details, please see the attached pdf document (which contains carefully formatted equations).

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#### Solution Summary

The solution method uses a recursion relation and the Bessel functions J_{1/2} and J_{-1/2} in conjunction with a proof by induction.