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(1) Consider the Gamma function I: RR, the Euler Gamma function restricted to real numbers: Show that: T(z) = e-tt-1dt 0 a) For every Є R, we have I (x + 1) = x(x); b) For every n Є N, we have I (n + 1) = n!, in particular F(1) = 1; c) F (})=√√T; d) I(-n): = ∞, for every n Є N, in particular I (0) = ∞; e) г(z)г(1 z) = - = sin πz for every z Є C. (2) Knowing that show that (m) k = m! (m-k)!k!! 0, if m > k, if m

(1) Consider the Gamma function I: RR, the Euler Gamma function restricted to real numbers: Show that: T(z) = e-tt-1dt 0 a) For every Є R, we have I (x + 1) = x(x); b) For every n Є N, we have I (n + 1) = n!, in particular F(1) = 1; c) F (})=√√T; d) I(-n): = ∞, for every n Є N, in particular I (0) = ∞; e) г(z)г(1 z) = - = sin πz for every z Є C. (2) Knowing that show that (m) k = m! (m-k)!k!! 0, if m > k, if m

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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Gamma Function

(1) Consider the Gamma function I: RR, the Euler Gamma function restricted to real numbers: Show that: T(z) = e-tt-1dt 0 a) For every Є R, we have I (x + 1) = x(x); b) For every n Є N, we have I (n + 1) = n!, in particular F(1) = 1; c) F (})=√√T; d) I(-n): = ∞, for every n Є N, in particular I (0) = ∞; e) г(z)г(1 z) = - = sin πz for every z Є C. (2) Knowing that show that (m) k = m! (m-k)!k!! 0, if m > k, if m <k, n = (-1)"T(n+B) n!T (B)
Transcribed Image Text:(1) Consider the Gamma function I: RR, the Euler Gamma function restricted to real numbers: Show that: T(z) = e-tt-1dt 0 a) For every Є R, we have I (x + 1) = x(x); b) For every n Є N, we have I (n + 1) = n!, in particular F(1) = 1; c) F (})=√√T; d) I(-n): = ∞, for every n Є N, in particular I (0) = ∞; e) г(z)г(1 z) = - = sin πz for every z Є C. (2) Knowing that show that (m) k = m! (m-k)!k!! 0, if m > k, if m <k, n = (-1)"T(n+B) n!T (B)
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