# Computer convolution of sech squared function

I try to compute the analytic form of the convolution of $\operatorname{sech}$ squares function. Mathematica takes so much time to try and find the answer. Here is my code:

Simplify[Convolve[A*Sech[1.7627*x/\[Sigma]]^2, A*Sech[1.7627*x/\[Sigma]]^2, x, y]]


Convolve[A Sech[α x/σ]^2, A Sech[α x/σ]^2, x, y, Assumptions -> {α, σ} \[Element] Reals]

• Since only the ratio \[Alpha]/\[Sigma] appears, you can significantly speed up the calculation by replacing the ratio with a single variable and then returning the ratio. – Bob Hanlon Apr 4 '17 at 16:37
• To do both: Module[{a, expr}, expr = A Sech[α x/σ]^2 /. α -> a*σ; Assuming[{a \[Element] Reals}, Simplify[ Convolve[expr, expr, x, y] /. a -> α/σ]]] // AbsoluteTiming – Bob Hanlon Apr 4 '17 at 17:00