A property of the _digamma_ function On the last paragraph of the article on Wikipedia about the digamma function I found 1 _The digamma function appears in the regularization of divergent integrals_ $$ \int _{0}^{\infty }{\frac {dx}{x+a}} $$ _this integral can be approximated by a divergent general Harmonic series, but the following value can be attached to the series_ $$\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n+a}}=-\psi (a) $$ There is no reference, I'd like to know where the last divergent sum value comes from. thanks in advance

The idea of a regularization is to subtract some (constant) infinity to be left with a finite expression possibly meaning something. In this case, the relevant formula is mentioned in the article at Wikipedia: $$\psi (a)=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+a}}\right).$$ In other words, we add the constant infinity (divergent series) $$\gamma -\sum _{n=0}^{\infty }\frac {1}{n+1}$$ to get the hopefully meaningful function $-\psi(a).$