Order of Integration (calculus) - Basic Theorems

Basic Theorems

A good discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T.W. Körner. He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example:

Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:

Theorem I Let f(x, y) be a continuous function of constant sign defined for a ≤ x < ∞, c ≤ y < ∞, and let the integrals

and regarded as functions of the corresponding parameter be, respectively, continuous for c ≤ y < ∞, a ≤ x < ∞. Then if at least one of the iterated integrals and converges, the other integral also converges and their values coincide.

Theorem II Let f(x, y) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integrals

and be respectively, uniformly convergent on every finite interval c ≤ y < C and on every finite interval a ≤ x < A. Then if at least one of the iterated integrals and converges, the iterated integrals and also converge and their values are equal.

The most important theorem for the applications is quoted from Protter and Morrey:

Suppose F is a region given by   where p and q are continuous and p(x) ≤ q(x) for a ≤ x ≤ b. Suppose that f(x, y) is continuous on F. Then

The corresponding result holds if the closed region F has the representation   where r(y) ≤ s(y) for c ≤ y ≤ d.  In such a case,

In other words, both iterated integrals, when computable, are equal to the double integral and therefore equal to each other.