Digamma at n+1/2

Theorem

The following formula holds for $n=1,2,3,\ldots$: $$\psi \left( n +\dfrac{1}{2} \right) = -\gamma - 2 \log(2)+ 2 \left( 1 + \dfrac{1}{3} + \ldots + \dfrac{1}{2n-1} \right),$$ where $\psi$ denotes the digamma function and $\gamma$ denotes the Euler-Mascheroni constant.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.3.4$