I have to get the value of k in this equation: $\frac{(\lambda T)^k [ln(\lambda T)-\psi(k+1)]}{\Gamma(k+1)}=0$, where $\psi$ is the digamma function. Since $\Gamma(k+1)$ is in the denominator and the value for k would be $-\infty$ for $(\lambda T)^k$ I am trying to solve this: $ln(\lambda T)=\psi(k+1)$. I've tried to solve replace $\psi(k+1)$ by its intregral representation: 
But I haven´t been able to get an answer.