Need help with $\int_{0}^\infty x^n\sin(bx)e^{-ax^2}\, dx$

Question

I need help with this integral:

$$\int_{0}^\infty x^n \sin(bx)e^{-ax^2}\, dx$$ where $a, b \in \Bbb R$ and $n\in \Bbb N$.

Mathematica wasn't very helpful.

Answer 1

Mathematica wasn't very helpful.

That's because $~\sin bx=\Im(e^{ibx}),~$ whereas the other exponential has $x^2$ in its exponent,

which is what ultimately made a simple solution in terms of the $\Gamma$ function impossible.

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Let $~I(a,b)~=~\displaystyle\int_0^\infty\sin(bx)~e^{-ax^2}~dx,~$ $~J(a,b)~=~\displaystyle\int_0^\infty\cos(bx)~e^{-ax^2}~dx,~$ evaluate both

in closed form, using Gaussian integrals and error functions, then take full advantage of

the fact that $n\in\mathbb N.~$ Notice that, depending on the parity of n, the initial integral can be

written as $~\dfrac{d^k}{da^k}~I(a,b),~$ for $n=2k,~$ or as $~\dfrac d{db}~\dfrac{d^k}{da^k}~J(a,b),~$ for $n=2k+1.$