How to prove: $\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(\alpha_1+x)}{\beta_1^{\alpha_1+x}}\, \frac{\Gamma(\alpha_2-x)}{\beta_2^{\alpha_2-x}}\, dx=\frac{\Gamma(\alpha_1+\alpha_2)}{(\beta_1+\beta_2)^{\alpha_1+\alpha_2}} \qquad \text{Re}(\alpha_1),\text{Re}(\alpha_2),\text{Re}(\beta_1),\text{Re}(\beta_2)>0$ Does anyone can make the proof easier in the link provide below in the comment?
please help with the a gamma function since i don't even have the idea?
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complex-analysis
special-functions
gamma-function
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0Ask the ghost of Ramanujan - this is just the ty$p$e of $r$esult that he found. – 2011-10-08