(a) If Bym is the mth zero of (d/dp)J,(BvmP/a), show that the Bessel functions are orthogonal over the interval [0, a] with an orthogonality integral | 1. (Bum2) J» (Bun²) o dp = 0, ) Ju O = 0, m#n, v>-1. (b) Derive the corresponding normalization integral (m = n). v2 [J,(Bym)]², v> -1. ANS. (b)

(a) If Bym is the mth zero of (d/dp)J,(BvmP/a), show that the Bessel functions are orthogonal over the interval [0, a] with an orthogonality integral | 1. (Bum2) J» (Bun²) o dp = 0, ) Ju O = 0, m#n, v>-1. (b) Derive the corresponding normalization integral (m = n). v2 [J,(Bym)]², v> -1. ANS. (b)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(a) If Bym is the mth zero of (d/dp)J,(BvmP/a), show that the Bessel functions are
orthogonal over the interval [0, a] with an orthogonality integral
| 1. (Bum2) J» (Bun²) o dp = 0,
) Ju
O = 0, m#n, v>-1.
(b) Derive the corresponding normalization integral (m = n).
v2
[J,(Bym)]², v> -1.
ANS. (b)

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