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For complex variables y, z = C, show that Jn(y + 2) = Σ Jm(Y)Jn-m(2). M=18

For complex variables y, z = C, show that Jn(y + 2) = Σ Jm(Y)Jn-m(2). M=18

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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help me with (3) please

 

please use Laurent series, be detailed. thank you

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" . 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" . 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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