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?
2026-04-08 12:33:07.1775651587
please help with the a gamma function since i don't even have the idea?
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