Advanced Calculus fi..

456 Advanced Calculus, Fifth EditionOne sees at once that if I-", f (x) dx exists in the usual sense, then0lirn la f (x) dx and :li la f (x) dxa+coboth exist, and hence the principal value exists and equals the usual improper integral.However, the principal value may exist even though the usual value does not. Forexample,1 1 +(a - 1 P =,= lirn - loga-.a,2 1 + ( ~ + 1 ) ~Here the integral from 0 to oo is +oo, from -oo to 0 is -oo. There is in effect acancellation of the two infinities. This is related to a certain symmetry of the graphof the function.The concept of principal value can be extended to the integral of a function ffrom a to b, where f is discontinuous only at c, with a < c < b. Here one defines:For example,[LC-'f (x) dx = lim€'O+f (x) dx + Jb (x) dx] .c+cHere the integrals from 0 to 1 and from 1 to 3 are +oo and -oo, respectively, andagain there is a cancellation of the two infinities, made clear in the evaluation of thelast limit. It is easily seen, as previously, that whenever the integral of f from a tob exists, as usual, as an improper integral, the principal value exists and equals theprevious value; the example just given shows that the principal value can exist evenwhen the usual improper integral does not exist. Of course, the principal value canalso fail to exist, as the following example shows:[[y dx = lim f dx +c+O+1;i dx]= a+O+ Iim [(: + 1 ) + (-1 + f)] =Here there is no cancellation because the function is always positive.If f is continuous for -00 c x c oo except at c, then one can use a principalvalue both for c and for large x as follows:03 c-E(P) / f (x) dx = lirn lirn- wf(x)dx +Sa f(x)dx].c+cThis procedure can be adapted to the case of several discontinuities cl , . . . , c,.