The integral \(\frac{1}{2\pi i}\oint\frac{\exp\left[\frac{z}{2}\left(t-\frac{1}{t}\right)\right]}{t^{n+1}}\,dt\), where \(n\) is an integer and the closed contour is a circle enclosing the origin, can be used to :
generate Bessel’s functions of first kind
The function \(g(z,t)=\exp\left[\frac{z}{2}\left(t-\frac{1}{t}\right)\right]\) is the generating function of the Bessel functions of the first kind: \(g(z,t)=\sum_{n=-\infty}^{\infty}J_{n}(z)\,t^{n}\), a Laurent series in \(t\) about \(t=0\).
By the Laurent coefficient formula, the coefficient of \(t^{n}\) is \(\frac{1}{2\pi i}\oint\frac{g(z,t)}{t^{n+1}}dt\) around any circle enclosing the origin. Hence the given integral equals \(J_{n}(z)\).
The other families have different generating functions: Hermite \(e^{2xt-t^{2}}\), Legendre \((1-2xt+t^{2})^{-1/2}\), and associated Legendre functions are obtained from the Legendre generating function by differentiation, none of which matches this exponential.
Check: setting \(n=0\) and expanding to lowest order in \(z\) gives \(J_{0}(z)\approx 1-\frac{z^{2}}{4}\), the known series start of the Bessel function of order zero.
Hence, generate Bessel’s functions of first kind.
Similar Questions
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Important Questions from Mixed Topic (MH-SET)
.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
