4. The generating function for the regular Bessel functions of integer order is [(-)] = [(x). I exp Using this fact to show that m=18 2Jm(x) = Jm-1(x) — Jm+1(x). Although this only shows the relation to hold for integer m values, it turns out to hold for non-integer m values as well.
4. The generating function for the regular Bessel functions of integer order is [(-)] = [(x). I exp Using this fact to show that m=18 2Jm(x) = Jm-1(x) — Jm+1(x). Although this only shows the relation to hold for integer m values, it turns out to hold for non-integer m values as well.
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 4.
![**Generating Function for Regular Bessel Functions**
The generating function for the regular Bessel functions of integer order is given by:
\[
\exp\left[\frac{x}{2}\left(h - \frac{1}{h}\right)\right] = \sum_{m=-\infty}^{\infty} h^m J_m(x).
\]
Using this fact, we can show that:
\[
2J'_m(x) = J_{m-1}(x) - J_{m+1}(x).
\]
Although this relation is shown to hold for integer \( m \) values, it also applies to non-integer \( m \) values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2F5beb7e1c-35a4-493b-be5d-4dc25cbc7e69%2Fuvkitnj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Generating Function for Regular Bessel Functions**
The generating function for the regular Bessel functions of integer order is given by:
\[
\exp\left[\frac{x}{2}\left(h - \frac{1}{h}\right)\right] = \sum_{m=-\infty}^{\infty} h^m J_m(x).
\]
Using this fact, we can show that:
\[
2J'_m(x) = J_{m-1}(x) - J_{m+1}(x).
\]
Although this relation is shown to hold for integer \( m \) values, it also applies to non-integer \( m \) values.
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