(1) Use the transformation u = y- z to rewrite I as an integral over a semi-infinite rectangular domain. Hence show that I = 0 if n is an odd integer. (ii) Assuming n is an even integer, use another substitution to determine I(a, n) in terms of the beta function.
(1) Use the transformation u = y- z to rewrite I as an integral over a semi-infinite rectangular domain. Hence show that I = 0 if n is an odd integer. (ii) Assuming n is an even integer, use another substitution to determine I(a, n) in terms of the beta function.
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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Transcribed Image Text:(b) Let A denote the semi-infinite strip described by x –1<ysr+1 and a 2 0. Consider the integral
I(a, n) = 2"(y – 2)"
A (1+ x)20+2
dz dy,
where a > -1 is a real number and n is a positive integer.
(i) Use the transformation u = y - z to rewrite I as an integral over a semi-infinite rectangular domain. Hence show that I = O if n is an odd integer.
(ii) Assuming n is an even integer, use another substitution to determine I(a, n) in terms of the beta function.
(i) Relate your answer to the gamma function, and thus evaluate I(3/2, n) for even integers n, simplifying your answer as much as possible.
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