Functions Defined by Integrals; Improper Integrals

Abstract

The solutions of problems in differential equations, especially those which arise in physics and engineering, are frequently given in terms of integrals. Most often either the integrand of the integral representing the solution is unbounded or the domain of integration is an unbounded set. In this chapter we develop rules for deciding when it is possible to interchange the processes of differentiation and integration—commonly known as differentiation under the integral sign. When the integrand becomes infinite at one or more points or when the interval of integration is infinite, a study of the convergence of the integral is needed in order to determine whether or not the differentiation process is allowable. We establish the required theorems for bounded functions and domains in this section and treat the unbounded case in Sections 11.2 and 11.3.