Suppose that $F: \mathbb{C} \to \mathbb{C}$ is analytic when $\operatorname{Re}(z)>0$. Assume that it is possible to show that $F$ can be represented as a Laplace transform $$ F(z) = \int_0^{ + \infty } {\rm e}^{ - zt} f(t)\,{\rm d}t, $$ for $\operatorname{Re}(z)>a$ with some non-negative constant $a$ and $f$ being analytic in a neighbourhood of the positive real line. Does it follow from the analyticity of $F$ for $\operatorname{Re}(z)>0$ that $a=0$?
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I love how you asked thus question 2 years ago and never touched it ever again. – Rounak Sarkar Oct 12 '21 at 07:35
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2@RounakSarkar What do you expect from me to do with it? – Gary Oct 12 '21 at 07:52
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If I don't get an answer to one of my questions then I actively seek it by giving bounties and cleverly editing the question to bring to more attention. I made the previous comment because I didn't expected such inactivity in a question from a big user like you. – Rounak Sarkar Oct 12 '21 at 08:26
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@Gary Have you come across an answer to this question? – bob Mar 22 '23 at 05:53
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@bob No, unfortunately I have not. – Gary Mar 22 '23 at 06:41