I'm attempting to evaluate the following integral, so far, with little success. Any help would be appreciated:
$ \ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx $
All paramaters are real.
Mx
I'm attempting to evaluate the following integral, so far, with little success. Any help would be appreciated:
$ \ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx $
All paramaters are real.
Mx
Combining summand is the exponents one has that the integral is a Laplace transform $x\to g$ of the function $f(x,a,h)=e^{-a x^2} x^k \sin hx\;$. For $k=0$, $a,g>0\;$ Mma gives $ L[f]= -\frac{i \sqrt{\pi } e^{\frac{(g-i h)^2}{4 a}} \left(\text{erfc}\left(\frac{g-i h}{2 \sqrt{a}}\right)-e^{\frac{i g h}{a}} \text{erfc}\left(\frac{g+i h}{2 \sqrt{a}}\right)\right)}{4 \sqrt{a}}. $ Now note that for natural $k$ the result is equal to $(-1)^k\frac{\partial^k}{\partial g^k}L[f]$, so it seems where is no good formula for arbitrary $k$.