Integration similaring to Fourier transform of Gaussian function

Question

I would like to calculate the integral:

$$\int^{\infty}_{0}x\cdot \exp(-x^2)\cdot \exp(-ikx)dx$$

Are there some tricks to solve it?

Many thanks.

Answer 1

You can complete the square in the exponent, and get an answer in terms of the Dawson function:

$$\frac{1}{4} \left(2-\sqrt{\pi } i k e^{\frac{i^2 k^2}{4}} \text{erfc}\left(\frac{i k}{2}\right)\right).$$

Answer 2

\begin{align} \int_{0}^{\infty}xe^{-x^2}e^{-ikx}dx & = -\frac{1}{2}\int_{0}^{\infty}e^{-ikx}\frac{d}{dx}e^{-x^2}dx \\ & = -\left.\frac{1}{2}e^{-ikx}e^{-x^2}\right|_{0}^{\infty}-\frac{ik}{2}\int_{0}^{\infty}e^{-x^2}e^{-ikx}dx \\ & = \frac{1}{2}-\frac{ik}{4}\sqrt{2\pi}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2}e^{-ikx}dx \end{align} So the problem reduces to the Fourier transform of a Gaussian.