The Logarithmic Transform of a Polynomial Function Expressed in Terms of the Lerch Function

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.


Introduction
Perusing the current literature, we noticed there is not many articles related to a Logarithmic transform applied to a polynomial function. The intention here is to assist researchers by adding another set of mathematical tools for which to further current research in the areas requiring such transforms. The Logarithmic transform proposed by Reynolds and Stauffer [1] is used in this work and applied to a form of the Euler integral of the first kind and expressed in terms of the Lerch function.
In the present work, the authors used their contour integral method and applied it to a special case of the Beta function in [2] to derive a definite integral and expressed its closed form in terms of a special function. This derived integral formula was then used to provide formal derivations in terms of special functions and fundamental constants and summarized in a Table. The Lerch function being a special function has the fundamental property of analytic continuation, which enables us to widen the range of evaluation for the parameters involved in our definite integral. The Lerch function is a special function that generalizes the Hurwitz zeta function, the polylogarithms, and so many interesting and important special functions. The definite integral derived in this manuscript is given by: where the parameters k, a, n, m are general complex numbers with −1 < Re(m) < 1 and −1 < Im(n) < 1. The derivation of the definite integral follows the method used by us in [1] which involves Cauchy's integral formula. The generalized Cauchy's integral formula is given by: where C is in general an open contour in the complex plane where the bilinear concomitant [1] has the same value at the end points of the contour. The method in [1] involves using a form of Equation (2) then multiply both sides by a function, then take a definite integral of both sides. This yields a definite integral in terms of a contour integral. A second contour integral is derived by multiplying Equation (2) by a function and performing some substitutions so that the contour integrals are the same.

Definite Integral of the Contour Integral
We use the method in [1]. The variable of integration in the contour integral is z = w + m. The cut and contour are in the second quadrant of the complex z-plane. The cut approaches the origin from the interior of the second quadrant and the contour goes round the origin with a zero radius and is on opposite sides of the cut. Using Equation (2) we replace y by log(a x −n−1 − 1 ) then multiply by x n x −n−1 − 1 m . Next we take the finite integral over x ∈ [0, 1] to get: from Equation (3.249.7) in [2] where −1 < Re(w + m) < 1. The logarithmic function is given for example in Section 4.1.2 in [3]. We are able to switch the order of integration over w + m and x using Fubini's theorem since the integrand is of a bounded measure over the space C × [0, 1]. The steps used to derive the final line in Equation (3) are as follows.
Firstly, using Equation (3.249.7) set v = u to get the definite integral in terms of the cosecant function. Next replace u → 1/u to get the definite integral: Next setting u = −u, we will iterate this integral by looking at the following forms: 1 0 . . .

Derivation of the First Contour
In this section we will again use Cauchy's integral Formula (2) and take the infinite sum to derive equivalent sum representations for the contour integrals. We proceed using Equation (2) and replace y by log(a) + iπ(2y + 1) and multiply both sides by − 2iπme iπm(2y+1) n+1 and take the infinite sum over y ∈ [0, ∞) simplifying in terms of the Lerch function to get: where csch(ix) = −i csc(x) from Equation (4.5.10) in [3] and Im(w + m) > 0 for the sum to converge.

Derivation of the Second Contour
Using (11) we replace k → k − 1 and simplify to get: [3] and Im(w + m) > 0 for the sum to converge.

Definite Integral in Terms of the Lerch Function
Proof. Since the right-hand side of Equation (3) is equal to the sum of the right-hand sides of Equations (11) and (12) we can equate the left-hand sides and simplify the Gamma function to get the stated result.
Proof. Use Equation (13) and set k = 0 and simplify using entry (2) in the Table below (64:12:7) in [4]. Note this integral for n, a general complex number is not defined in Wolfram Mathematica. This is a simple example of the Beta function, see Equation (3.249.7) in [2].

Proposition 17.
1 0 Proof. Use Equation (13) and replace a → e a , k → −1. Next form a second equation by replacing a → −a and take their difference and set n = 1.
Proof. Use Equation (40) to take the first partial derivative with respect to k and apply L'Hopitals' rule as k → 0 and simplify.
Proof. Use (44) take the first derivative with respect to k then apply L'Hopital's rule as k → 0 set a = 1 and use Equation (64:10:2) in [4].

Discussion
In this present work, we used our contour integral method to derive the logarithmic transform in terms of the Lerch function. We then used this new integral relation to derive new definite integrals in terms of the other special functions and fundamental constants. A table of definite integrals summarizing our results was produced for easy reading.

Conclusions
In this paper, we derived a method for expressing definite integrals in terms of special functions using our contour integration method. The contour we used was specific to solving integral representations in terms of the Lerch function. The present results in this work can be extended to fractional calculus and fractal calculus [6,7]. There is also a possible connection to the variational iteration method and the homotopy perturbation method [8]. We expect that other contours and integrals can be derived using this method for future work. The results presented were numerically verified for both real and imaginary values of the parameters in the integrals using Mathematica by Wolfram.