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How can I prove that in formal way, this function doesn't have inverse Laplace transform. $$ F(s)=\frac{\sin(s)}{\sqrt{s}} $$ Strictly it should be in Bromwich contour method. Could you please tell me how to represent in poles in the contour.

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  • $\begingroup$ But unable to found poles this function. Any help from anyone it will be great help. $\endgroup$
    – meli0das
    Commented Jun 13, 2021 at 5:40
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    $\begingroup$ curiously enough, Mathematica does return an inverse Laplace transform of $s^{-1/2}\sin s$, but a numerical check shows it's mistaken. $\endgroup$
    – Carlo Beenakker
    Commented Jun 13, 2021 at 7:00
  • $\begingroup$ But I have to give a holistic approach to prove this ! But how ? $\endgroup$
    – meli0das
    Commented Jun 13, 2021 at 10:31
  • $\begingroup$ Mathematica says: $$-\frac{1}{\sqrt{2 \pi } \left(t^2+1\right)^{3/4} \sqrt{\frac{t}{\sqrt{t^2+1}}+1}}$$ $\endgroup$
    – Mariusz Iwaniuk
    Commented Sep 11, 2023 at 9:47

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