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From hyperbolic to complex Euler integrals

N. M. Belousov, G. A. Sarkissian, V. P. Spiridonov

Abstract

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

From hyperbolic to complex Euler integrals

Abstract

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

Paper Structure

This paper contains 21 sections, 14 theorems, 284 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Properties of the gamma functions
  3. Complex rational case
  4. Hyperbolic case
  5. From hyperbolic to complex rational beta integral
  6. Complex limits of the hyperbolic gamma function
  7. Outline
  8. Interchanging limits
  9. Calculating the $\delta$-limit
  10. From hyperbolic to complex rational conical function
  11. Conical functions
  12. Outline
  13. Interchanging limits
  14. Calculating the $\delta$-limit
  15. From hyperbolic to complex rational hypergeometry
  16. ...and 6 more sections

Key Result

Theorem 1

Assume that the parameters of the function $\mathcal{I}(z) \equiv \mathcal{I}(z; \lambda, g, \omega_1, \omega_2)$ satisfy the conditions and Then the following limit holds $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Types of gamma functions: solid arrows represent limits, dashed arrows correspond to algebraic connections
  • Figure 2: Poles (black) and zeros (white) of hyperbolic gamma function
  • Figure 3: Poles of $\mathcal{I}(z)$ with $\delta = 1$ and $\delta = 0.2$

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 19 more