Integral to Infinity of Function over Argument

Theorem
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le A$.

Let $\laptrans f = F$ denote the Laplace transform of $f$.

Then:
 * $\displaystyle \int_0^\infty {\dfrac {\map f t} t} = \int_0^{\to \infty} \map F u \rd u$

provided the integrals converge.

Also see

 * Laplace Transform of Integral