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(21 Show that Jn(2) = Σ m=0 (−1)m (3)n+2m m!(n +m)!

(21 Show that Jn(2) = Σ m=0 (−1)m (3)n+2m m!(n +m)!

Advanced Engineering Mathematics
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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there is only one quesiton i need help. the website says its rejected because i submit multiple quesitns. :< i dont understand why 

help me with part (2) please

please be detailed. you need to use Laurent series, and in the end i think we will end up with two sums, and then they will combine to the final answer.

For complex variables z, w E C and integer n & Z, the function Jn (z) is defined as the Laurent coefficient of wn in the following Laurent expansion, ∞ f(w) := exp ¹ ( 1⁄ ² (w − ²¹ ) ) = Σ In (2) w" . wn. ω n=1x The functions Jn (z) are also known as Bessel functions of the first kind. They play an important role in wave propagation phenomena as they appear when looking for separable solutions to the Laplace equation and the Helmholtz equation. Show that 1 CTT Jn (z) == "* cos(nº — z sin 6) de . 0 (Hint: compute the Laurent series of f around the point w = 0 and use the symmetry of the integrand to simplify.) Conclude that Jn (x)| ≤1 for all real x € R. (2) Show that (-1)m()+2 m!(n + m)! m=0 (3) For complex variables y, z = C, show that Jm(y)Jn-m(2). Jn (z) = Jn(y + 2) = Σ m=-∞
Transcribed Image Text:For complex variables z, w E C and integer n & Z, the function Jn (z) is defined as the Laurent coefficient of wn in the following Laurent expansion, ∞ f(w) := exp ¹ ( 1⁄ ² (w − ²¹ ) ) = Σ In (2) w" . wn. ω n=1x The functions Jn (z) are also known as Bessel functions of the first kind. They play an important role in wave propagation phenomena as they appear when looking for separable solutions to the Laplace equation and the Helmholtz equation. Show that 1 CTT Jn (z) == "* cos(nº — z sin 6) de . 0 (Hint: compute the Laurent series of f around the point w = 0 and use the symmetry of the integrand to simplify.) Conclude that Jn (x)| ≤1 for all real x € R. (2) Show that (-1)m()+2 m!(n + m)! m=0 (3) For complex variables y, z = C, show that Jm(y)Jn-m(2). Jn (z) = Jn(y + 2) = Σ m=-∞
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