Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions

Abstract

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 ∫ 0 ∞ log ( 1 ± e − α y ) R ( k , a , y ) d y in terms of a special function, where R ( k , a , y ) is a general function and k, a and α are arbitrary complex numbers, where R e ( α ) > 0 .

An integral's journey over the real line

“…We use the method in [8]. Here the contour is similar to Figure 2 in [8]. Using a generalization of Cauchy's integral formula we first replace y by log(y) followed by multiplying both sides by log(1 − zy)…”

“…The derivations follow the method used by us in [8]. The generalized Cauchy's integral formula is given by…”

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

“…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)…”

“…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)…”

Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function

“…In our case the constants in the formulas are general complex numbers subject to the restrictions given below. The derivations follow the method used by us in [3][4][5]. The generalized Cauchy's integral formula is given by…”