We derive the infinite sum of the Lerch function in terms of the incomplete gamma function and the Lerch function. Special cases are evaluated in terms of fundamentals constants. All the results in this work are new.

We present a method using contour integration to evaluate the definite integral of the form ${\int }_0^{\infty }{log}^k\left(ay\right)R\left(y\right)dy$ in terms of special functions, where $R\left(y\right)=\frac{y^m}{1+\alpha y^n}$ and $k,m,a,\alpha$ and n are arbitrary complex numbers. We use this method for evaluation as well as to derive some interesting related material and check entries in tables of integrals.

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ${\int }_0^{\infty }log(1\pm e^{-\alpha y})R(k,a,y)dy$ in terms of a special function, where $R(k,a,y)$ is a general function and k, a and $\alpha$ are arbitrary complex numbers, where $Re\left(\alpha \right)>0$ .

We present a method using contour integration to evaluate the definite integral of arctangent reciprocal logarithmic integrals in terms of infinite sums. In a similar manner, we evaluate the definite integral involving the polylogarithmic function $L{i}_k\left(y\right)$ in terms of special functions. In various cases, these generalizations give the value of known mathematical constants such as Catalan’s constant G, Aprey’s constant $\zeta \left(3\right)$ , the Glaisher–Kinkelin constant A, $log\left(2\right)$ , and $\pi$ .