# Hand-to-hand combat with thousand-digit Integrals

David H. Bailey

### Abstract

Jonathan Borwein's premier contribution to modern mathematics is his uncanny ability to identify and pursue lines of research that (a) are mathematically exciting, (b) involve computation in an essential way, and (c) are widely accessible and appealing to the younger generation (and even to the public at large). His work is the ultimate answer to those who claim that "real mathematicians don't compute".

One of the most common themes of research that he and I have jointly pursued, which is very much in keeping with Jon's down-to-earth experimental approach, is the analytic evaluation of definite integrals by means of experimental computations. In particular, we have succeeded in analytically identifying many thousands of specific definite integrals, mostly originating in the field of mathematical physics, by means of high-precision numerical computations, coupled with intelligent searches for analytic formulas using Ferguson's PSLQ integer relation detection algorithm. These studies have required both substantial computational ingenuity (in computing numerical values of these integrals to high precision) and substantial 6mathematical ingenuity (in finding formal proofs of the resulting experimental identities). This talk will summarize some recent work in this field, including results on integrals that arise in the fields of quantum field theory and random walks.

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