Parametric Differentiation Revisited

Abstract

A number of now-ancient texts in advanced calculus employed the method of parametric differentiation to deduce evaluations for variety of complicated definite integrals (e.g., [4,7]). In his autobiography [3], R. P. Feynman mentioned how he frequently used this approach when confronted with difficult integrations associated with mathematical and physical problems. He referred to that method as a different box of tools. Most modern texts either ignore the subject or provide only one or two examples to illustrate theorems on the uniform convergence of improper integrals. (See [8] for more recent treatment of integrals.) However, the method of parametric differentiation has much broader scope than this. It can, in fact, be introduced into the elementary calculus to simplify the treatment of partial fractions with repeated factors, to carry out integrations ordinarily handled by parts, and to treat integrals that use substitutions. When tied in with other mathematical machinery (e.g., complex variables), it can be used to deduce formulas for special functions and to solve problems in differential equations that involve iterated operators. Finally, the method can be conveniently used in conjunction with mathematical software packages that handle symbolic differentiations. In this note, we give three examples from calculus to illustrate the flexibility of this approach at the elementary level. One of the integrals treated is connected with the t-distribution. We also note, without providing complete discussion, how this technique can be used to evaluate the Riemann zeta function at 2p, p = 1, 2,.. ., and how it can be used in differential equations. A more extensive set of applications is available from the author. In the work to follow, we call upon the Leibniz product rule for differentiation as well as theorems that permit interchanging orders of differentiation and integration (see [5, 6]). We also need to make frequent use of the formula