Can one explicitly compute the Laplace transform of a function involving the reciprocal of the gamma function? See the picture for the Laplace transform. Thank you very much.
Dear Feng, I have used Mathematica built-in function LaplaceTransform but no explicit expression has been computed, sorry. Gianluca
Try power series. 1/Gamma(z) is an entire function with no poles and a series expansion for the whole complex plane. See Whittaker & Watson CUP (1920)
Surely there is not an elementary form. Indeed the gamma symbol encloses several approximations to the gamma function, depending on computer package. Only a "small" number of functions fullfills such requirement. Now, your concern here is whether the proposed series converges quick enough to be used in the numerical applications you are looking for.
Ok, you could look at the techniques used to solve both integrals and employ them with that proposed by you. May be your integral has some similarities with them to the light of the integration tricks they dealt with. Are they from Gradshteyn and Ryzhik or Abramovitz and Stegun?Or from another handbook?
Fine, well done. I should need some time to derive them and look for similarities with the previous one, sometimes that helps but it is not (similarity in form, or resemblance), a guarantee at all that it will lead you to similar solution, or any.
If I remember some trick or get a better idea, I'll write you.
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