Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a

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Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

(-1)* ο0 Σ 2k Jo(x) 22k (k!) )? k=0

a. Write out the first four terms of J0.

b. Find the radius and interval of convergence of the power series for J0.

c. Differentiate J0 twice and show (by keeping terms through x6) that J0 satisfies the equation x2y"(x) + xy'(x) + x2y(x) = 0.

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Calculus Early Transcendentals

ISBN: 978-0321947345

2nd edition

Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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