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While trying to evaluate the Laplace transform below $$I = \int_{0}^{\infty}e^{-st}B(\frac{1}{2}-it,\frac{3}{2}+it)\mathrm{d}t,$$

invoking

LaplaceTransform[Beta[0.5-it, 1.5+it], t, s],

yields

$$\frac{B(\frac{1}{2}-it,\frac{3}{2}+it)}{s},$$

which is quite weird since the output of a Laplace transform has to be an exclusive function of $s$, but this output still includes $t$. What am I missing here?

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  • $\begingroup$ Looks like a bug, you shouldn't have s in output..I came across similar bugs, the solution was to rewrite integral using Feynam trick $\endgroup$ Commented Feb 6 at 1:39
  • $\begingroup$ @YaroslavBulatov: $s$ is OK. One should not have $t$ in the output. $\endgroup$
    – User
    Commented Feb 6 at 1:42
  • $\begingroup$ There were couple of bugs I ran across in laplace.transform, my workaround is using ilaplaceFeynmann and ilaplaceMellin helpers from mathematica.stackexchange.com/q/286895/217 $\endgroup$ Commented Feb 6 at 1:43
  • $\begingroup$ @YaroslavBulatov: The definitions of these transforms are apparently not the same as that ofLaplaceTransform. So, how do their results help here? $\endgroup$
    – User
    Commented Feb 6 at 1:46
  • 3
    $\begingroup$ The Beta function does not include t, only the variable it. What did you intend? i*t or I*t? $\endgroup$
    – Bob Hanlon
    Commented Feb 6 at 2:27

1 Answer 1

2
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$Version

(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)

Clear["Global`*"]

Mathematica appears to be able to handle imaginary values for i

LaplaceTransform[Beta[1/2 - I*t, 3/2 + I*t] //
   FunctionExpand, t, s] // FullSimplify

(* (1/(8 π))(-2 π PolyGamma[0, (π + s)/(4 π)] + 
  2 π PolyGamma[0, 1/4 (3 + s/π)] + 
  I (PolyGamma[1, (π + s)/(4 π)] - PolyGamma[1, 1/4 (3 + s/π)])) *)

Letting i == a*I,

f[i_, s_] = ((LaplaceTransform[(Beta[1/2 - i*t, 3/2 + i*t] /. i -> a*I) //
        FunctionExpand, t, s] // FullSimplify) /. a -> i/I) // FullSimplify

(* (1/(8 i π))(-2 I π (PolyGamma[0, 1/4 + (I s)/(4 i π)] - 
     PolyGamma[0, 3/4 + (I s)/(4 i π)]) - 
  PolyGamma[1, 1/4 (1 + (I s)/(i π))] + 
  PolyGamma[1, 1/4 (3 + (I s)/(i π))]) *)
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