Double Integral of Logarithm and Exponential Function Expressed in terms of the Lerch Function

This paper contains new explicit computations of some integrals containing elementary functions, such as powers, logarithms, exponentials. In this work the authors use their contour integral method to derive a double connected modified Bessel function second kind $K_{\nu}(z)$ and express it in terms Lerch function. A table pairs is given for interested readers. The majority results are new.