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    Bessel functions

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    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).

    © BrainMass Inc. brainmass.com March 5, 2021, 12:13 am ad1c9bdddf
    https://brainmass.com/math/number-theory/bessel-functions-491847

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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.

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