ABSTRACT
By restricting the way limits are taken in a definite integral, a convergent expression may sometimes be obtained.
suppose that f(x) is integrable everywhere in the interval, except at the point z. By definition (see page 53), the integral J: f(x)dx has the value
(24.1)
where the limits are to be evaluated independently. If the value in (24.1) does not exist, but the value
lim {1•-'Y f(x) dx + 1" f(x) dx} ,_o o •+7 (24.2) does exist, then this value is the principal value, or the Cauchy principal value, of the improper integral I. This is denoted I= f: f(x)dx. Using the relation f: = f: +f:, we can define the principal value of an integral with multiple singularities.
