[Paper Review] Generalized Elliptic Integrals and Applications

Nikos Bagis|arXiv (Cornell University)|Apr 4, 2013

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper derives closed-form evaluations of generalized elliptic integrals and Rogers-Ramanujan continued fraction (RRCF) integrals using hypergeometric functions, modular forms, and special values of the Dedekind eta function. It establishes novel connections between RRCF, hypergeometric series, and elliptic integrals, yielding exact results for integrals involving $ f(-q)^4 q^{-5/6} R(q)^{5 u} $ and $ u(q)^n $, with applications to modular equations and Ramanujan-type $ /pi $ formulas.

ABSTRACT

We use some general properties, presented in previous work, to evaluate special cases of integrals relating Rogers-Ramanujan continued fraction, eta function and elliptic integrals.

Motivation & Objective

• To evaluate special cases of integrals involving the Rogers-Ramanujan continued fraction (RRCF), Dedekind eta function, and elliptic integrals using advanced special functions.
• To derive closed-form expressions for integrals of the form $ \int_0^{R^{-1}(\sqrt[5]{(-11+5\sqrt{5})/2})} f(-q)^4 q^{-5/6} R(q)^{5\nu} dq $ using hypergeometric and gamma functions.
• To establish new modular equations and transformations for RRCF and eta function integrals via substitution and Appell $ F_1 $ hypergeometric identities.
• To construct Ramanujan-type $ \pi $-formulas of arbitrary precision using solutions to modular equations involving $ m(R) $ and $ \psi^*(x) $.
• To generalize the evaluation of integrals involving $ u(q) = R(q)^{-5} - 11 - R(q)^5 $, linking them to elliptic integrals and known special functions.

Proposed method

• Uses Ramanujan’s identities for the Rogers-Ramanujan continued fraction (RRCF), including $ \frac{1}{R(q)} - 1 - R(q) = \frac{f(-q^{1/5})}{q^{1/5} f(-q^5)} $ and $ \frac{1}{R^5(q)} - 11 - R^5(q) = \frac{f(-q)^6}{q f(-q^5)^6} $, to relate RRCF to modular forms.
• Applies the derivative formula $ R'(q) = 5^{-1} q^{-5/6} f(-q)^4 R(q) \sqrt[6]{R(q)^{-5} - 11 - R(q)^5} $ to transform integrals into rational forms in $ R(q) $.
• Employs the Appell $ F_1 $ hypergeometric function and the Euler integral representation of the Gauss hypergeometric function to evaluate integrals of the form $ \int x^m (ax^2 + bx + c)^n dx $.
• Uses the transformation $ x \to w^{1/5} $, $ w \to y/\rho_2 $, and $ w^{-1} - 11 - w \to t $ to reduce complex integrals to standard hypergeometric forms.
• Relies on known relations between the eta function, modular forms, and complete elliptic integrals, such as $ f(-q) = 2^{1/3} \pi^{-1/2} q^{-1/24} k^{1/12} k'^{1/3} K(k)^{1/2} $, to express integrals in terms of $ k $, the singular modulus.
• Applies the differential relation $ \frac{dq}{dk} = \frac{-q \pi^2}{2 k k'^2 K(k)^2} $ to derive $ \frac{dR}{dk} $, enabling integration in terms of $ k $ and $ F_1 $ or $ {}_2F_1 $ functions.

Experimental results

Research questions

• RQ1 Can integrals of the form $ \int_0^{R^{-1}(\sqrt[5]{(-11+5\sqrt{5})/2})} f(-q)^4 q^{-5/6} R(q)^{5\nu} dq $ be evaluated in closed form using hypergeometric functions?
• RQ2 How can the Rogers-Ramanujan continued fraction and Dedekind eta function be used to derive new modular equations and transformations?
• RQ3 What is the exact evaluation of $ \int_0^{u^{-1}(x)} u(q)^n \frac{f(-q^5)^5}{f(-q)} dq $ for $ n \in \{1/2, 1, 3/2, 2, \dots\} $, and how does it relate to elliptic integrals?
• RQ4 Can Ramanujan-type $ \pi $-formulas of arbitrary precision be constructed from solutions to modular equations involving $ m(R) $ and $ \psi^*(x) $?
• RQ5 What is the connection between the generalized beta function $ B_\alpha(x) $, the solution $ \beta_r $ to $ B_\alpha(1 - \beta_r)/B_\alpha(\beta_r) = \sqrt{r} $, and hypergeometric series expansions?

Key findings

• The integral $ \int_0^{R^{-1}(\sqrt[5]{(-11+5\sqrt{5})/2})} f(-q)^4 q^{-5/6} R(q)^{5/2} dq $ evaluates exactly to $ \frac{27\Gamma(5/6)\Gamma(8/3)\sin\left(\frac{2}{3}\arctan\left(\sqrt{\frac{-11+5\sqrt{5}}{11+5\sqrt{5}}}\right)\right)}{2\sqrt{5\pi}} $, a closed-form expression in terms of gamma and trigonometric functions.
• For $ u(q) = R(q)^{-5} - 11 - R(q)^5 $, the integral $ \int_0^{u^{-1}(x)} u(q)^n \frac{f(-q^5)^5}{f(-q)} dq = -\int_0^x \frac{t^{n-1}}{\sqrt{125 + 22t + t^2}} dt $, which reduces to elliptic integrals when $ n \in \{1/2, 1, 3/2, \dots\} $.
• The identity $ \pi \int_{\sqrt{r}}^{\infty} \eta(it/2)^4 dt = 3\sqrt[3]{2k_r} \cdot {}_2F_1\left[\frac{1}{3}, \frac{1}{6}; \frac{7}{6}; k_r^2\right] = 5 \int_0^{R(q)} \frac{dx}{x \sqrt[6]{x^{-5} - 11 - x^5}} $ links the eta function to hypergeometric functions and RRCF integrals.
• For $ r = 4 $, the evaluation $ R(e^{-2\pi}) = -\frac{1+\sqrt{5}}{2} + \sqrt{\frac{5+\sqrt{5}}{2}} $ leads to $ \int_2^{\infty} \eta(it/2)^4 dt = 5\pi^{-1} \int_0^{R(e^{-2\pi})} \frac{dx}{x \sqrt[6]{x^{-5} - 11 - x^5}} $, a new exact formula.
• The function $ m(R) = \sin^2\left(\frac{\pi}{2(R+1)}\right) $ satisfies the modular equation $ m(R+1) = \frac{1 - \sqrt{1 - m(R/2)}}{2} $, enabling recursive construction of Ramanujan-type $ \pi $-formulas.
• The series $ \sum_{n=0}^\infty \frac{(1/4)_n (3/4)_{-n}}{(1/2)_{-n} n!} \frac{m(R)^{n+1/2}}{n+1/2} = 2\arcsin(\sqrt{m(R)}) $, and when $ m(R) $ is evaluated via radicals, it yields a Ramanujan-type $ \pi $-formula of arbitrary precision.