Need help with singular integral?

Question

I need to evaluate a definite integral from minus infinity to plus infinity. The integrand (as a function of the variable p) is the product of F(p) and exp(i*p*x), where (p) is the integration variable, (x) is a given Real constant, and (i) is the usual sqrt(-1). The function F(p) is equal to (p/sqrt(p*p + m*m)), where (m) is another given (Real) constant.

As p goes to (+/-) infinity, F(p) approaches to (+/-)1, and the integral diverges in the strict sense, but it may converge to something like the Dirac-delta function, because the same integral with F(p) replaced by 1 everywhere indeed converges to 2*pi*delta(x), the Dirac-delta function. In fact, this integral may converge to a continuous function because F(p) is odd in p relative to p=0.

Can you help me with this integral?

Answer 1

The integral seems to be ambiguously defined. Lim C approaches infinity of integral from -C to +C is *one* way to disambiguate it. Beware though, the integrand violates the conditions assumed for a Fourier transform, so the later's theorems and inverse may not apply, if that is your plan.