The Mellin Transform of Logarithmic and Rational Quotient Function in terms of the Lerch Function

<jats:p>Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by <jats:inline-formula>
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                  </jats:inline-formula>, where the parameters <jats:inline-formula>
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                  </jats:inline-formula> are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.</jats:p>


Introduction
e well-known book on Mellin transforms by Brychkov et al. [1] contains a vast number of integral formulae. In this work, we focus on adding to section (2.5.1) featuring "logarithmic and algebraic functions." In particular, this work will derive a closed form solution for the integral given by ∞ 0 x m− 1 log k (ax) in terms of the Lerch function. is is achieved by applying our contour integration method to an integral transform from Erdeyli's book [2] to yield a definite integral in terms of the Lerch function. We have also produced other formulae in terms of the Lerch function using our method in [3]. We use this new integral transform in terms of the Lerch function to derive new formulae for Brychkov's book [1] thereby extending the current tables in section (2.5.1). We will also derive new Malmsten-type log(log(x)) integrals in terms of special functions and provide new formula also extending the work done by Malmsten [4]. e derivations of the formulae in this work follow the method used by us in [5]. is method involves using the generalized Cauchy's integral formula given by and multiplying both sides of (2) by a function, followed by taking a definite integral on both sides which yields a definite integral in terms of a contour integral. en, we multiply both sides of equation (2) by another function and take the infinite sum of both sides such that the contour integral of both equations are the same. Here, C is, in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour.

Definite Integral of the Contour Integral
We use the method in [5]. e variable of integration in the contour integral is α � m + w. e cut and contour are in the first quadrant of the complex α-plane with 0 < Re(α) < u. e cut approaches the origin from the interior of the first quadrant and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy's integral formula, we first replace y by log(ax) followed by multiplying both sides by and then taking the definite integral with respect to from equation (3.263) in [6], where − 1 < Re(m + w) < 2 and Re(β) > 0, |arg(c)| < π. We are able to switch the order of integration over w and x using Fubini's theorem since the integrand is of bounded measure over the space C × [0, ∞).

Derivation of the First Contour Integral.
In this section, we will derive the Lerch function representation for the first contour integral given by Using equation (2) and replacing y by log(a) + log(β) + (1/2)iπ(2y + 1) and then multiplying both sides by − iπβ m e (1/2)iπm(2y+1) we obtain We then take the infinite sum over y ∈ [0, ∞) simplifying in terms of the Lerch function to obtain 2 Journal of Mathematics from (1.232.3) in [6], where Im(m + w) > 0 in order for the sum to converge, and csch(ix) � − i csc(x) from (4.5.10) in [8].

Derivation of the Second Contour Integral.
In this section, we will derive the Lerch function representation for the second contour integral given by Using equation (2) and replacing y by log(a) + log(c) + iπ(2y + 1) and then multiplying both sides by we obtain We then take the infinite sum over y ∈ [0, ∞) simplifying in terms of the Lerch function to obtain Journal of Mathematics 3 from (1.232.3) in [6], where Im(m + w) > 0 in order for the sum to converge, and csch(ix) � − i csc(x) from (4.5.10) in [8].

Derivation of the ird Contour Integral.
In this section, we will derive the Lerch function representation for the third contour integral given by Using equation (2) and replacing y by log(a) + log(β) + (1/2)iπ(2y + 1) and then multiplying both sides by we obtain We then take the infinite sum over y ∈ [0, ∞) simplifying in terms of the Lerch function to obtain  [6], where Im(m + w) > 0 in order for the sum to converge, and sech(ix) � i sec(x) from (4.5.11) in [8].

Definite Integral in terms of the Lerch Function
Proof. Since the right-hand side of equation (4) is equal to the sum of equations (10), (13), and (18), we may equate the left-hand sides to get the stated result.
□ Note when m is replaced by m − 1, we get the Mellin transform representation.

Derivation of an Entry in Gradshteyn and Ryzhik
In this section, we will derive equation (3.263) in [6]. Using equation (19) and setting k � 0 and simplifying, we obtain from entry (2) below

Definite Integral in terms of the Hypergeometric Function
In this section, we will look at derivations in terms of the Hypergeometric function which can be derived from the Lerch function and apply it to get special cases. We will make use of equation (9.559) in [6] and Journal of Mathematics 7.1. Entry H1. Using (21) and setting a � − 1, β � 1, c � 1, and m � 1/2 and simplifying, we obtain Next, we rationalize the denominator on the left-hand side and simplify to obtain 7.3. Entry H3. Using equation (21), taking the first partial derivative with respect to m, and setting β � c � 1, a � e πi , and m � − 1/2 simplifying, we obtain after rationalizing the real and imaginary parts.

Entry H4.
Using equation (21), taking the first partial derivative with respect to m, and setting β � c � 1, a � e πi/2 , and m � − 1/2 simplifying, we obtain after rationalizing the real and imaginary parts.

Definite Integrals of log(log(x)) with a Cubic
Denominator. In this section, we will use equation (19) to derive a Table of Malmsten's type log(log(x)) integrals [4] in terms of the Log gamma function log(Γ(x)) and the fundamental constant π. e forms, which we will derive, are given by and we use equations (1.10.10) in [10], (64 : 13 : 3) in [9], and (24.14.2) in [11] for simplifying the formula derived.
8.9. Entry M8. Using equation (33) and applying L'Hopital's rule to the right-hand side as c ⟶ − 1 and simplifying, we obtain 8.10. Entry M9. Using equation (33) and setting c � 2 and simplifying, we obtain 8.11. Entry M10. Using equation (33) and setting c � − 2 and simplifying, we obtain 8.12. Entry M11. Using equation (33), then taking the first partial derivative with respect to c, and setting c � 1 and simplifying, we obtain 8.13. Entry M12. Using equation (33), then taking the first partial derivative with respect to c, and setting c � − 1 and simplifying, we obtain 8.14. Entry M13. Using equation (33), then taking the second partial derivative with respect to c, and setting c � 1 and simplifying, we obtain ∞ 0 log(log(x)) (x − 1)(x + 1) 4 dx � − 28ζ(3) + 3π 4 − iπ 3 (29 + log(64)) + 40π 2 (c + log(2/π)) 96π 2 . (40) 8.15. Entry M14. Using equation (33), then taking the second partial derivative with respect to c, and setting c � − 1 and simplifying, we obtain 8.16. Definite Integrals of log(log(x)) and a Rational Quotient Function. In this section, we again look at Malmsten integrals [4] of the form Using equation (19) and formulating a second equation by replacing m by n, then taking the difference followed by taking the first partial derivative with respect to k, and then setting k � 0 and simplifying in terms of the derivative of the Lerch function, we obtain 2iπe iπn c n Φ ′ e 2iπn , 0, ((π − i log(c))/2π)

Conclusion
In this article, we derived a table of definite integrals which can be included in current books such as [1,2,6]. We found that using our simultaneous contour integral method, we were able to generate very interesting new integral formulae.
We also looked at analytic continuity and principle value methods to evaluate integrals when singularities were present. One of the consequences of this work is the production of a table of definite integrals which researchers may use to advance their work if they find any of these formulae applicable. We will be looking at other integrals using this contour integral method for future work. e results presented were numerically verified for both real and imaginary values of the parameters in the integrals using Mathematica by Wolfram.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.