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I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula:

a.) Prove that

$$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = 2\int_0^{1/2}x^{p-1}(1-x)^{p-1}\mathrm dx$$

b.) Make a suitable change of variable (a) and derive the duplication formula for the Gamma function:

$$\Gamma (2p)\Gamma\left(\frac12\right) = 2^{2p-1}\Gamma (p)\Gamma\left(p-\frac12\right)$$

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    Start here: http://en.wikipedia.org/wiki/Beta_function#Relationship_between_gamma_function_and_beta_function2011-04-23

2 Answers 2