Question

Prove that $\int_{0}^{\infty }\frac{x^{a-3/2}dx}{[ x^2+( b^2-2)x+1]^a}=b^{1-2a}\frac{\Gamma(1/2)\Gamma(a-1/2)}{\Gamma(a)}$

score 385 · Answer 1 · 2013-02-01 02:07:42

385

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Prove that $\int_{0}^{\infty }\frac{x^{a-3/2}dx}{[ x^2+( b^2-2)x+1]^a}=b^{1-2a}\frac{\Gamma(1/2)\Gamma(a-1/2)}{\Gamma(a)}$

Published on 01 Feb 2013 - 2:07

score 389 · Answer 2 · 2013-02-07 22:44:32

389

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Published on 07 Feb 2013 - 22:44

score 278 · Answer 3 · 2013-02-13 14:02:04

278

Views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

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score 98 · Answer 4 · 2013-02-21 10:13:30

98

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9.5k

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score 518 · Answer 6 · 2013-02-26 13:14:36

518

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score 159 · Answer 7 · 2013-02-28 23:42:38

159

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1.6k

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857

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An integral related to the beta function: $\int_{-1}^{1} \frac{(1+x)^{2m-1}(1-x)^{2n-1}}{(1+x^{2})^{m+n}} \, dx $

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score 410 · Answer 10 · 2013-03-03 01:53:07

410

Views

Closed form of, or series for $\int_{\epsilon-i\infty}^{\epsilon+i\infty}\frac{e^{az+b^2z^2}}{\sin\pi z}\,dz$

Published on 03 Mar 2013 - 1:53

score 907 · Answer 11 · 2026-04-02 10:34:04

907

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score 1.9k · Answer 13 · 2013-03-15 11:28:40

1.9k

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score 665 · Answer 14 · 2013-03-17 19:04:54

665

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score 313 · Answer 15 · 2013-03-19 14:00:11

313

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Strategy for Improper Integrals Related to the Beta Function 2

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List Question

Prove that $\int_{0}^{\infty }\frac{x^{a-3/2}dx}{[ x^2+( b^2-2)x+1]^a}=b^{1-2a}\frac{\Gamma(1/2)\Gamma(a-1/2)}{\Gamma(a)}$

Whether is my conclusion about euler gamma function correct? is it new conclusion $\Gamma (0)=i\pi$?

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Is there any meaning on $\displaystyle\Gamma(i)$

Help evaluating a gamma function

Integral $\int\limits_0^\infty \prod\limits_{k=0}^\infty\frac{1+\frac{x^2}{(b+1+k)^2}}{1+\frac{x^2}{(a+k)^2}} \ dx$

Calculating an almost Gamma integral

how to calculate gamma function in programming language?

An integral related to the beta function: $\int_{-1}^{1} \frac{(1+x)^{2m-1}(1-x)^{2n-1}}{(1+x^{2})^{m+n}} \, dx $

Closed form of, or series for $\int_{\epsilon-i\infty}^{\epsilon+i\infty}\frac{e^{az+b^2z^2}}{\sin\pi z}\,dz$

Understanding a particular evaluation of $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$

Summation of Gamma functions and fractions

Interesting Integral $\int_{-\infty}^{\infty}\frac{e^{i nx}}{\Gamma(\alpha+x) \Gamma(\beta -x)}dx$

Similar function to gamma distribution

Strategy for Improper Integrals Related to the Beta Function 2

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