The following equation arises while solving the Hamiltonian expectation energy for my quantum dot calculations and I plan to use variational approach to minimize the energy ($\beta$ is the variational parameter). $$\int_0^{100} \frac{e^{-2\beta[\rho^2 + (z-z_i)^2]^{1/2}}}{[\rho^2 +(z-z_i)^2]^{1/2}}dz$$ where $\rho = 50$ and $z_i$ will be varied from 0 to 20.
question: is there a way this integral could be solved either analytically/numerically (say even for a simple case when $z_i$ = 0)?
Thanks for the help