Complete p-elliptic integrals and a computation formula of $$\pi _p$$ π p for $$p=4$$ p = 4

Abstract

The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function sin p θ and its half-period π p . It is shown, only for p = 4, that the generalized p-elliptic integrals yield a computation formula of π p in terms of the arithmetic-geometric mean. This is a π p -version of the celebrated formula of π, independently proved by Salamin and Brent in 1976.

Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function

“…In [1], properties of the extended generalized Mittag-Leffler function are studied in details, and it is given that E γ,δ,k,c μ,α,l (t; p) is absolutely convergent for k < δ + R(μ). For other recent results see [15,37,38,46]. Let S be the sum of series of absolute terms of E γ,δ,k,c μ,α,l (t; p), then we have E γ,δ,k,c μ,α,l (t; p) ≤ S. We will use this property of extended generalized Mittag-Leffler function in sequel.…”

Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind

“…In the past few centuries, the complete elliptic integrals of the first and second kinds K(r) and E(r), defined on [ have been found that they have many important applications in mathematics as well as in physics and engineering, including the evaluation of the length of curves [1,5,9,10,22,28,39,42,46,55], the algorithm of the circumference ration π [11,13,21,41], the computations of electromagnetic field and the study of the period of the simple pendulum [12,15,19,25,30]. In 1990s, the complete elliptic integrals K(r) and E(r) appeared in geometric function theory frequently, especially in conformal and quasiconformal mappings [3,4,6,17,27,31,38,54,59].…”

On a function involving generalized complete (p, q)-elliptic integrals

“…is known as the generalized sine function with two parameters p, q > 1 in the literature (see, [9,10,14,21,24,[28][29][30]), and defined as the inverse function of…”