101. The gamma function, which plays an important role in advanced applications, is defined for n > 1 by I(n) = | t"-le+ dt (a) Show that the integral defining r(n) converges for n > 1 (it actually converges for all n > 0). Hint: Show that t"-'e- 1 is an integer. Hint: Use (b) repeatedly. Thus, I'(n) provides a way of defining n-factorial when n is not an integer.
101. The gamma function, which plays an important role in advanced applications, is defined for n > 1 by I(n) = | t"-le+ dt (a) Show that the integral defining r(n) converges for n > 1 (it actually converges for all n > 0). Hint: Show that t"-'e- 1 is an integer. Hint: Use (b) repeatedly. Thus, I'(n) provides a way of defining n-factorial when n is not an integer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:101. The gamma function, which plays an important role in advanced applications, is defined for n > 1 by
I(n) = | t"-le+ dt
(a) Show that the integral defining r(n) converges for n > 1 (it actually converges for all n > 0). Hint:
Show that t"-'e-<t² for t sufficiently large.
(b) Show that I'(n +1) = nI(n) using Integration by Parts.
(c) Show that I'(n + 1) = n! if n > 1 is an integer. Hint: Use (b) repeatedly. Thus, I'(n) provides a way of
defining n-factorial when n is not an integer.
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