THE CAUCHY-SCHL ¨ OMILCH TRANSFORMATION T. AMDEBERHAN, M. L. GLASSER, M. C. JONES, V. H. MOLL, R. POSEY, AND D. VARELA Abstract. The Cauchy-Schl¨ omilch transformation states that for a function f and a, b > 0, the integral of f (x 2 ) and af ((ax - bx -1 ) 2 over the interval [0, ∞) are the same. This elementary result is used to evaluate many non- elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given. 1. Introduction The problem of analytic evaluations of definite integrals has been of interest to scientists for a long time. The central question can be stated as follows: given a class of functions F and an interval [a,b] ⊂ R, express the integral of f ∈ F I =  b a f (x) dx, in terms of special values of functions in an enlarged class G. Many methods for the evaluation of definite integrals have been developed since the early stages of Integral Calculus, which resulted in a variety of ad-hoc techniques for producing closed-form expressions. Although a general procedure applicable to all integrals is undoubtedly unattainable, it is within reason to expect a systematic cataloguing procedure for large groups of definite integrals. To this effect, one of the authors has instituted a project to verify all the entries in the popular table by I. S. Gradshteyn and I. M. Ryzhik [13]. The website http://www.math.tulane.edu/∼vhm/web − html/pap-index.html contains a series of papers as a treatment to the above-alluded project. Naturally, any document containing a large number of entries, such as the table [13] or the encyclopedic treatise [20], is likely to contain errors, many of which arising from transcription from other tables. The earliest extensive table of integrals still accessible is [2], compiled by Bierens de Haan who also presented in [3] a survey of the methods employed in the verification of the entries from [13]. These tables form the main source for [13]. The revision of integral tables is nothing new. C. F. Lindman [17] compiled a long list of errors from the table by Bierens de Haan [4]. The editors of [13] maintain the webpage http://www.mathtable.com/gr/ Date : May 12, 2009. 1991 Mathematics Subject Classification. Primary 33. 1