[Paper Review] Integrals in Gradshteyn and Ryzhik: Hyperbolic and trigonometric function

Mark W. Coffey|arXiv (Cornell University)|Feb 22, 2018

Analytic Number Theory Research12 references3 citations

TL;DR

This paper provides rigorous proofs and extensions for definite integrals involving hyperbolic and trigonometric functions from Gradshteyn and Ryzhik's table, using series expansions and connections to the Hurwitz zeta and polygamma functions. A key contribution is a general integral formula involving $\int_0^\infty \frac{\cosh(bx)}{\sinh(ax)} x^{p-1} dx$ expressed via the Hurwitz zeta function, which unifies and extends multiple entries in sections 4.118–4.124.

ABSTRACT

The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.

Motivation & Objective

• To rigorously prove and extend selected definite integrals involving hyperbolic and trigonometric functions from Gradshteyn and Ryzhik's table.
• To establish connections between these integrals and special functions such as the Hurwitz zeta, Gamma, and polygamma functions.
• To identify and correct potential errors in existing table entries, particularly in section 4.124.
• To derive new integrals arising from the study of elliptic functions and their Laurent expansions in terms of Hurwitz numbers.

Proposed method

• Employing geometric series expansions of $\sinh^{-1}(ax)$ and $\cosh^{-1}(ax)$ to evaluate integrals involving $\cosh(bx)/\sinh(ax)$.
• Using the integral representation of the Hurwitz zeta function $\zeta(s,a) = \sum_{n=0}^\infty (n+a)^{-s}$ for $\text{Re}(s) > 1$ to express results.
• Applying the functional equation $\zeta(s,a+1) = \zeta(s,a) - a^{-s}$ and known reductions to the Riemann zeta function.
• Leveraging the relation $\psi^{(n)}(x) = (-1)^{n+1} n! \zeta(n+1,x)$ to connect integrals to polygamma functions.
• Using analytic continuation and differentiation under the integral sign to derive equivalent forms of known entries.
• Deriving new integrals from properties of the Weierstrass $\wp$-function and its Laurent expansion involving Hurwitz numbers.

Experimental results

Research questions

• RQ1 How can the integrals in Gradshteyn and Ryzhik's table involving hyperbolic and trigonometric functions be rigorously proven using special function identities?
• RQ2 What is the general form of the integral $\int_0^\infty \frac{\cosh(bx)}{\sinh(ax)} x^{p-1} dx$ and how does it relate to the Hurwitz zeta function?
• RQ3 Are there errors in the cited entries of Gradshteyn and Ryzhik, particularly in section 4.124, and if so, how can they be corrected?
• RQ4 Can new integrals be derived from the number-theoretic properties of Hurwitz numbers arising in elliptic function expansions?

Key findings

• A general formula is derived: $\int_0^\infty \frac{\cosh(bx)}{\sinh(ax)} x^{p-1} dx = \frac{\Gamma(p)}{(2a)^p} \left[ \zeta\left(p, \frac{a-b}{2a}\right) + \zeta\left(p, \frac{a+b}{2a}\right) \right] $ for $\text{Re}(p) > 1$ and $|\text{Re}(b)| < |\text{Re}(a)|$.
• The result recovers known entries such as 3.523.1 and 3.524.5, and establishes their equivalence through analytic continuation and differentiation.
• For $p=2$, the integral evaluates to $\frac{\pi^2}{4a^2} \sec^2\left(\frac{\pi b}{2a}\right)$, confirming entry 4.111.6 via the trigamma reflection formula.
• A corrected version of entry 4.124.1 is proposed, with evidence suggesting citation reversal between 4.124.1 and 4.124.2 in the source material.
• New integrals are derived from the Laurent expansion of the Weierstrass $\wp$-function with periods $\tilde{\omega}$ and $\tilde{\omega}i$, involving Hurwitz numbers.
• Specific new results include $\int_0^\infty \frac{(\cos t + 1)}{\cosh t - \cos t} \left[ \sinh(t/2) - \sin(t/2) \right] dt = 1 - \frac{\pi}{4}$ and $\int_0^\infty \frac{(\cos t + 1)t}{\cosh t - \cos t} \left[ \cosh(t/2) - \cos(t/2) \right] dt = \tilde{\omega}^2 - 4$, where $\tilde{\omega} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(1/4)}{\Gamma(3/4)}$.