I have seen some papers that refer the following lemma.
Lemma. Let be a nonnegative continuous function on the half-line , and assume that the absissa of convergence of the Laplace transform of is finite.
Then, has a singularity at the point . More precisely, .
It is said that the lemma can be found in the reference following, but I could not find it.
- D. V. Widder. An Introduction To Transform Theory. Academic Press, Newyork, 1971.
I really need the proof of this lemma, and don't know where to find it.
It would be very appreciated if someone can help me due to this direction.
See that the statement colored with red is wrong
See the proof