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43. The gamma function is defined by T(@) e-'pa-1 dt, (a > 0). (a) Prove that F'(1) = 1. (b) Prove that T(@+1) = «T(æ). In fact, if n is a positive integer, show that I'(n + 1) = n!. (c) Show that Γ (α+ 1 ) L{t“}(s) = sa+1 Indeed, if n is a positive integer, use this result to show that L{t"}(s) = n!/s"+1.

43. The gamma function is defined by T(@) e-'pa-1 dt, (a > 0). (a) Prove that F'(1) = 1. (b) Prove that T(@+1) = «T(æ). In fact, if n is a positive integer, show that I'(n + 1) = n!. (c) Show that Γ (α+ 1 ) L{t“}(s) = sa+1 Indeed, if n is a positive integer, use this result to show that L{t"}(s) = n!/s"+1.

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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43. The gamma function is defined by [(@) = | e1ª-1 dt, (a > 0). Γ(α) - e-lta-1 dt, (a) Prove that IT(1) = 1. (b) Prove that I'(a+1) = «T(@). In fact, if n is a positive integer, show that I'(n + 1) = n!. (c) Show that Γ (α +1) L{t“}(s) = sa+1 Indeed, if n is a positive integer, use this result to show that L{t"}(s) = n!/s"+.
Transcribed Image Text:43. The gamma function is defined by [(@) = | e1ª-1 dt, (a > 0). Γ(α) - e-lta-1 dt, (a) Prove that IT(1) = 1. (b) Prove that I'(a+1) = «T(@). In fact, if n is a positive integer, show that I'(n + 1) = n!. (c) Show that Γ (α +1) L{t“}(s) = sa+1 Indeed, if n is a positive integer, use this result to show that L{t"}(s) = n!/s"+.
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