convolution product of $\sin(x)$ and $e^{-x^2}$

Question

I am supposed to find the convolution product of the two functions $f(x) = \sin(x)$ and $g(x) = e^{-x^2}$.

I tried different approaches (for example integrating by parts or calculating the Fourier transforms of the functions in order to multiply and back-transform it then), but I always end up with results that are “blowing up”. I am not sure whether I am making a mistake in those approaches or if there might be some kind of easier trick to solve this?

Thank you for your help!

Answer 1

Maple says $$ \int_{-\infty }^{\infty }\!\sin \left( x \right)\, {{\rm e}^{- \left( y- x \right) ^{2}}}\,{\rm d}x =\sqrt{\pi}\, e^{-1/4}\sin y $$

Answer 2

$$I=\int\,\sin \left( x \right)\, {{\rm e}^{- \left( y- x \right) ^{2}}}\,dx=\Im\int e^{i x-(x-y)^2}\,dx$$ $$(x-y)^2-ix=\left(x-y-\frac{i}{2}\right)^2-i y+\frac{1}{4}$$ Let $x=t+y+\frac{i}{2}$ to make $$I=\Im\Bigg[ e^{i y-\frac 14}\int e^{-t^2}\,dt\Bigg]=\frac{ \sin (y)}{\sqrt[4]{e}}\int e^{-t^2}\,dt$$