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DreamWeaver

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\(\displaystyle \Gamma(x) = \int_0^{\infty}t^{x-1}e^{-t}\, dt\)And the Digamma function:

\(\displaystyle \psi_0(x) = \frac{d}{dx}\log \Gamma(x) = \frac{\Gamma'(x)}{\Gamma(x)}\)Prove Dirichlet's integral representation for the Digamma function:\(\displaystyle \psi_0(x) = \int_0^{\infty} \frac{1}{z}\left( e^{-z} - \frac{1}{(1+z)^x} \right)\, dz\)

**Hint:**

\(\displaystyle \int_{0}^{\infty}\int_{1}^{q}e^{-tz}\, dt\, dz\)

in two different ways, and equate the results.