Gamma function limt to integral question
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I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,
$$
Gamma(x) = lim_{nrightarrow infty}frac{n!n^{x-1}}{x(x+1)cdots(x+n-1)}, qquad xneq 0,-1,-2,dots
$$
How can I connect this formula with the standard definition:
$$
Gamma(x) = int^infty_0 t^{x-1}e^{-t}dt
$$
gamma-function
asked Nov 19 at 8:19
RedPen
212112
add a comment |
up vote
0
down vote
favorite
I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,
$$
Gamma(x) = lim_{nrightarrow infty}frac{n!n^{x-1}}{x(x+1)cdots(x+n-1)}, qquad xneq 0,-1,-2,dots
$$
How can I connect this formula with the standard definition:
$$
Gamma(x) = int^infty_0 t^{x-1}e^{-t}dt
$$
gamma-function
asked Nov 19 at 8:19
RedPen
212112
This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,
$$
Gamma(x) = lim_{nrightarrow infty}frac{n!n^{x-1}}{x(x+1)cdots(x+n-1)}, qquad xneq 0,-1,-2,dots
$$
How can I connect this formula with the standard definition:
$$
Gamma(x) = int^infty_0 t^{x-1}e^{-t}dt
$$
gamma-function
asked Nov 19 at 8:19
RedPen
212112
I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,
$$
Gamma(x) = lim_{nrightarrow infty}frac{n!n^{x-1}}{x(x+1)cdots(x+n-1)}, qquad xneq 0,-1,-2,dots
$$
How can I connect this formula with the standard definition:
$$
Gamma(x) = int^infty_0 t^{x-1}e^{-t}dt
$$
gamma-function
gamma-function
asked Nov 19 at 8:19
RedPen
212112
asked Nov 19 at 8:19
RedPen
212112
asked Nov 19 at 8:19
RedPen
212112
asked Nov 19 at 8:19
RedPen
212112
asked Nov 19 at 8:19
RedPen
212112
212112
This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35
add a comment |
This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35
This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35
add a comment |
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This is standard text book material. Any book that deals with Gamma function in detail has a proof.
– Kavi Rama Murthy
Nov 19 at 8:25
This can be shown using the factorization of $frac 1{Gamma(z)}$, which can be derived from the functional equation $Gamma(z)Gamma(1-z)=frac{pi}{sin(pi z)}$.
– lEm
Nov 19 at 8:35