Functions aren't executed

Can't figure out why Mathematica doesn't want to execute built-in functions? What can be wrong?

Input:

Convolve[1/(x - t)^(3/2)*Exp[-gamma/(2 (x - t))], 1/t^(3/2)*Exp[-gamma/(2 t)], t, x]


Output:

Convolve[E^(-(gamma/(2 (-t + x))))/(-t + x)^(3/2), E^(-(gamma/(2 t)))/t^(3/2), t, x]


Input:

Integrate[(1/(x - t)^(3/2)*Exp[-gamma/(2 (x - t))])*(1/t^(3/2)*
Exp[-gamma/(2 t)]), t]


Output:

\[Integral]E^(-(gamma/(2 t)) - gamma/(2 (-t + x)))/(t^(3/2) (-t + x)^(3/2)) \[DifferentialD]t

• Likely, that means Mathematica doesn't know a closed-form solution, if it exists. – J. M. is away Apr 1 '18 at 13:17
• And when asking questions, post code rather than pictures of code so that people can copy and paste the code into a workbook. – Bob Hanlon Apr 1 '18 at 13:34
• @BobHanlon, thanks for advice, I have added source code too and will do it further. – Hasek Apr 1 '18 at 13:57

1 Answer

The problem that you having with your problem is that the convolution envelope does not converge well when approaching zero. The best way is tackle this is to add a step function. This way will force all envelope to be zero for all negative numbers. Inspecting the envelope with Gamma = 1 Plot[1/t^(3/2)*Exp[-1/(2 t)] UnitStep[t], {t, -2, 2}]

Follow up with the convolution

ClearAll["Global*"]

Convolve[1/(x - t)^(3/2)*Exp[-\[CapitalGamma]/(2 (x - t))],
1/t^(3/2)*Exp[-\[CapitalGamma]/(2 t)] UnitStep[t], t, x]
`

The result is $\frac{1}{\Gamma ^2}$