# Digamma difference formula proof

$$\psi(1-x)-\psi(x)=\pi \cot(\pi x)$$

$$\textit{proof}$$

We know by the reflection formula  that

$$\Gamma(x)\Gamma(1-x)=\pi \csc(\pi x)$$

Now differentiate both sides

$$\psi(x)\Gamma(x)\Gamma(1-x)-\psi(1-x)\Gamma(x)\Gamma(1-x)=-\pi^2 \csc(\pi x)\,\cot(\pi x)$$

Which can be simplified

$$\Gamma(x)\Gamma(1-x)\left(\psi(1-x)-\psi(x)\right)=\pi^2 \csc(\pi x)\,\cot(\pi x)$$

Further simplifications using ERF results in

$$\psi(1-x)-\psi(x)=\pi \cot(\pi x) \,$$

This entry was posted in Digamma function and tagged , , . Bookmark the permalink.