The following is from the lecture notes of a course that I am taking:
Here $\hat{f}$ represents the Laplace transform of some holomorphic function.
I am having trouble understanding the limit. My issue is with the relationship of $|p|$ and $R$. I know that we have $R< |p|$ however, I am not sure that we can even talk about the limit as $|p|$ as then how do I deal with the supremum explicitly?
I am probably missing something obvious but I can't seem to spot why the limit is true.
Question: Can someone help me see how one deals with the limit formally?