Lebl-Basic-Analysis-I · Problem 5.5.6

The Riemann Integral

Mathematics · improper-integrals-laplace-transform

Problem

Take

f : [0, \infty) \to \mathbb{R}

, Riemann integrable on every interval

[0, b]

, and such that there exist

M

a

, and

T

, such that

\left|f(t)\right| \leq M e^{at}

for all

t \geq T

. Show that the Laplace transform of

f

exists. That is, for every

s > a

the following integral converges:

F(s) \coloneqq \int_0^{\infty} f(t) e^{-st} \, dt.