For complex variables z, w E C and integer ne Z, the function Jn (2) is defined as the Laurent coefficient of wn in the following Laurent expansion, 1 ƒ(w) := exp(-/22 (w − ¹ )) = £ In (2)wo". Σ W n=18 The functions Jn (2) 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 Jn (z) 1 = 7 * cos(ne - z sino) de . (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 E R. Show that 8 Jn(2) = Σ (-1)m (2)n+2m m!(n + m)! m=0 (3) For complex variables y, z E C, show that 8 Jn (y + z) = ΣJm(y)Jn-m (2). m=1∞
For complex variables z, w E C and integer ne Z, the function Jn (2) is defined as the Laurent coefficient of wn in the following Laurent expansion, 1 ƒ(w) := exp(-/22 (w − ¹ )) = £ In (2)wo". Σ W n=18 The functions Jn (2) 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 Jn (z) 1 = 7 * cos(ne - z sino) de . (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 E R. Show that 8 Jn(2) = Σ (-1)m (2)n+2m m!(n + m)! m=0 (3) For complex variables y, z E C, show that 8 Jn (y + z) = ΣJm(y)Jn-m (2). m=1∞
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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use laurent series, please strictly follow the hint; help me with (3)
(if you answer before, please dont post the same answer.thank you)
Transcribed Image Text:For complex variables z, w E C and integer n & Z, the function Jn (2) is defined as the Laurent coefficient
of wn in the following Laurent expansion,
f(w) := exp ¹ ( 1⁄ ² (w − ² ) ) = 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)n+2
m!(n + m)!
m=0
(3) For complex variables y, z = C, show that
Jn(y+2) =
Jm(y)Jn-m(2).
Jn (z)
=
m=-∞
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