Integration using Schwinger Parametrization

Abstract

This paper draws attention to an underappreciated method for evaluating certain types of definite integrals. The method relies on a substitution in the Eulerian integral for the Legendre gamma function, and has become known in some quarters as a Schwinger parametrization. We present some examples to illustrate the utility of this technique in the hope that by doing so we may convince the reader that it makes a valuable addition to one’s integration toolkit.

MSC:

• 26A36
• 26A06

ACKNOWLEDGMENT

It is a pleasure to thank the editor and the anonymous referee(s) for their careful reading of the paper and their many apt and insightful suggestions over the course of several revisions that greatly improved the paper.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Notes

1 https://en.wikipedia.org/wiki/Stigler’s_law_of_eponymy

2 Excluding the poles at zero and the negative integers.

Additional information

Notes on contributors

David M. Bradley

DAVID M. BRADLEY received his Ph.D. in mathematics from the University of Illinois at Urbana-Champaign. His research focus is classical analysis and special functions, with applications to other branches of mathematics, including number theory, combinatorics, difference-differential equations, and statistics. Bradley’s work is known in several countries; he has published over 40 refereed articles and given more than 50 invited lectures on his research. A classically trained pianist (former lifetime) and self-taught 5-ball juggler, Bradley has been a member of the mathematics faculty at the University of Maine since 1998, where he served a 3-year term as Department Chair from 2009 to 2012.

Department of Mathematics & Statistics, University of Maine, 5752 Neville Hall, Orono, ME 04469

[email protected]

Albert Natian

ALBERT NATIAN has been a professor of mathematics at Los Angeles Valley College for over 30 years. A collector of sorts, Natian holds Master’s degrees in Pure Math, Applied Math, Mathematical Finance and Physics. His research interests include combinatorics, applied probability, and quantum mechanics. He enjoys mathematical puzzles, especially those involving probability. In his downtime Natian does conceptual and abstract pen and ink drawings (MathAfterMath.Net) that involve patterns and objects in uncommon and extreme spatial relationships.

Department of Mathematics, Los Angeles Valley College, 5800 Fulton Avenue, Valley Glen, CA, 91401

[email protected]

Seán M. Stewart

SEÁN M. STEWART received his Ph.D. in theoretical physics from the University of Wollongong. After briefly working in Australia, he for many years taught mathematics and physics to engineers in Kazakhstan and the United Arab Emirates. He now lectures at King Abdullah University of Science and Technology in Saudi Arabia. He has always found it hard to resist the challenge of a definite integral and is the author of How to Integrate It: A Practical Guide to Finding Elementary Integrals published by Cambridge University Press.

Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

[email protected]