Series and integrals of hypergeometric type have many applications in mathematical physics. Investigation of the corresponding special functions can therefore be relevant for various computations in theoretical models of reality. Bailey chains provide powerful tools for generating infinite sequences of summation or transformation formulas for hypergeometric type series. For a review of the corr...

We present a numerical method to invert a general incomplete elliptic integral with respect to its argument and/or amplitude. The method obtains a solution by bisection accelerated by the half argument formulas and the addition theorems to evaluate the incomplete elliptic integrals and Jacobian elliptic functions required in the course. If a faster execution is desirable at the cost of complexi...

A number of families of elliptic-type integrals have been studied recently due to their importance and potential for applications in some problems of radiation physics. The object of this work is to present a unified and generalized form of such elliptic-type integrals and to study its properties, including recurrence formulas and asymptotic expansion.

A general class of (multiple) hypergeometric type integrals associated with the Jacobi theta functions is defined. These integrals are related to theta hypergeometric series via the residue calculus. In the one variable case, theta function extensions of the Meijer function are obtained. A number of multiple generalizations of the elliptic beta integral associated with the root systems An and C...

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems An ...

Elliptic really does have a connection with literal geometric ellipses, removed by a few steps. Integrals expressing arc length of ellipses are less elementary than integrals for inverse trigonometric functions, and were named elliptic because of this. Abel and Jacobi discovered in 1827 that the inverse functions are doubly periodic (explained below). This chain of events caused these functions...

The solid angle subtended by a right circular cylinder at a point source located at an arbitrary position generally consists of a sum of two terms: that defined by the cylindrical surface (Ωcyl) and the other by either of the end circles (Ωcirc). We derive an expression for Ωcyl in terms of elliptic integrals of the first and third kinds and give similar expressions for Ωcirc using integrals of...

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be...