Definite integrals involving combinations of logarithmic functions of complicated arguments and powers expressed in terms of the Hurwitz zeta function

In this manuscript, the authors derive closed formula for definite integrals of combinations of logarithmic functions of complicated arguments and powers and express these integrals in terms of the Hurwitz zeta. These derivations are then expressed in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.


Introduction
In this manuscript the authors derive the definite integrals given by (1) 1 0 log (x m + 1) log k a x + log k (ax) x dx and (2) in terms of the Hurwitz zeta function, where the parameters k, a, m and n are general complex numbers.A summary of the results is given in a table of integrals for easy reading.This work is important because the authors were unable to find similar results in current literature.Tables of definite integrals provide a useful summary and reference for readers seeking such integrals for potential use in their research.This work looks at definite integrals of the hyperbolic arctangent function and the product of logarithmic functions with complicated arguments and powers.We use our simultaneous contour integration method to aid in our derivations of the closed forms solutions in terms of the Hurwitz zeta function, which provides analytic continuation of the results.
The derivations follow the method used by us in [8].The generalized Cauchy's integral formula is given by (3) e wy w k+1 dw.
This method involves using a form of equation ( 3) 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 ( 3) by a function and performing some substitutions so that the contour integrals are the same.

Derivation of the first contour integral
We use the method in [8].Here the contour is similar to Figure 2 in [8].Using a generalization of Cauchy's integral formula equation ( 3), we we will form two equations and add them together.For the first and second equations replace y by log(ax) and y by log(a/x) respectively.Next we add these equations followed by multiplying both sides by log(x m +1) x and taking the definite integral over x ∈ [0, 1] to get (4) from equation (4.293.10) in [2] and the integral is valid for a, m and k complex and −1 < Re(w) < 0.Where the logarithmic function is defined in equation (4.1.2) in [16] 3. Derivation of the second contour integral Using a generalization of Cauchy's integral formula equation (3), we we will form two equations and add them together.For the first and second equations replace y by log(ax) and y by log(a/x) respectively.Next we add these equations followed by multiplying both sides by log(1−x n ) x and taking the definite integral over x ∈ [0, 1] to get (5) from equation (4.293.7) in [2] where −1 < Re(w) < 0.

Derivation of the infinite sum of the first contour integral
Again, using the method in [8] and equation (3), we replace y by log(a)+ iπ(2y+1) m multiply both sides by −2πi, replace k by k + 1 and take the infinite sum of both sides over y ∈ [0, ∞) simplifying in terms the Hurwitz zeta function to get (6) from equation (1.232.3) in [2] where csch(ix) = i csc(x) from (4.5.10) in [16] and Im(w) > 0 for the convergence of the sum.We use equation (9.521.1) in [2] where ζ(s, u) is the Hurwitz zeta function.

Derivation of the infinite sum of the second contour integral
Again, using the method in [8] and equation ( 3), we replace y by log(a)+ 2iπ(y+1) n multiply both sides by −2πi, replace k by k + 1 and take the infinite sum of both sides over y ∈ [0, ∞) simplifying in terms the Hurwitz zeta function to get

Derivation of the additional contours
Again, using the method in [8] and equation ( 3), we replace y by log(a), k by k + 1 and multiply both sides by πi simplify to get (8) iπ log k+1 (a) Again, using the method in [8] and equation (3), we replace y by log(a), k by k + 2 and multiply both sides by −n simplify to get Again, using the method in [8] and equation ( 3), we replace y by log(a), k by k + 2 and multiply both sides by −m simplify to get

Derivation of the definite integrals in terms of the Hurwitz zeta function
Since the right-hand side of equations ( 4) and ( 5) are equal to the sum of the right-hand sides of equations ( 6), ( 7), ( 8), ( 9) and (10) we can equate the left-hand sides simplifying the factorials to get (11)

Derivation of logarithmic and hyperbolic tangent integrals in terms of the Hurwitz zeta function
Using equations ( 11) and ( 12) and taking their difference simplifying we get (13) Using equations ( 11) and ( 12) and adding them, then simplifying we get ( 14) 9. Derivation of logarithmic and hyperbolic arctangent integrals in terms of the zeta function Using equations ( 13) and ( 14) and setting a = 1 simplifying we get (15) and ( 16) from entry (2) in Table below (64:7) in [12].

Derivation of logarithmic and hyperbolic arctangent integrals in terms of the log gamma function
Using equations ( 13) and ( 14) replacing a by e ai and applying L'Hopital's rule to the right-hand side as k → −1 respectively, simplifying we get (17) from equation (64:10:2) in [12].

Derivation of logarithmic and hyperbolic arctangent integrals in terms of the Digamma function
Using equations ( 13) and ( 14) replacing a by e ai and applying L'Hopital's rule to the right-hand side as k → −2 respectively, simplifying we get x a 2 + log 2 (x) 12. Derivation of logarithmic and hyperbolic arctangent integrals in terms of fundamental constants and special functions In this section we will derive definite integrals in terms of special functions and fundamental constants such as Euler's constant (γ), Catalan's constant (C), Glaisher's constant (A) and π.This section showcases just a subset of the range of evaluations of these integral formula.

Derivation of definite integrals of the logarithmic function
Using equation (11) we take the first partial derivative with respect to m, then replace m by m + 1.Next we form a second equation by replacing m by p in the new equation.Then we take the difference of these two new equations simplifying to get (33) Repeating the steps above using equation (12) simplifying to get (34) 13.1.Some special cases.

Discussion
In this work we looked at deriving definite integrals of combinations of logarithmic functions of complicated arguments and powers and expressed them in terms of the Hurwitz zeta function.One of the interesting properties of these integrals is by adding them we were able to get the integral of the product of the hyperbolic arctangent function and the logarithmic function.We formally derived a few integrals in terms of fundamental constants and special functions.One of our goals we to supply a table for easy reading by researchers and to have these results added to existing textbooks.
The results presented were numerically verified for both real and imaginary values of the parameters in the integrals using Mathematica by Wolfram.We considered various ranges of these parameters for real, integer, negative and positive values.We compared the evaluation of the definite integral to the evaluated Special function and ensured agreement.

Conclusion
In this paper we used our method to evaluate definite integrals using the Lerch function.The contour we used was specific to solving integral representations in terms of the Lerch function.We expect that other contours and integrals can be derived using this method.

Acknowledgments
This paper is fully supported by the Natural Sciences and Engineering Research Council (NSERC) Grant No. 504070.