0
$\begingroup$

Let $x_0$ and $\sigma$ be constants. How do we evaluate the following?

$$ \large \int^{L}_{-L}e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos x \, \mathrm{d}x $$

I think I can solve that with integration by parts. But I'm confused how to calculate the exponential function if I choose it as $\dfrac{\mathrm{d}v}{\mathrm{d}x}$.

$\endgroup$
1
$\begingroup$

This is not an answer but it is too long for a comment

As Martín-Blas Pérez Pinilla commented, the antiderivative is far away to be simple. One way to approach it is to define $$I=\int e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos (x) dx$$ $$J=\int e^{-\frac{(x-x_0)^2}{2\sigma^2}}\sin (x) dx$$ $$K=I +i J=\int e^{ix-\frac{(x-x_0)^2}{2\sigma^2}} dx$$ Now, completing the square and integrating, $$K=-i \sqrt{\frac{\pi }{2}}\, \sigma \, e^{i x_0-\frac{\sigma ^2}{2}} \text{erfi}\left(\frac{\sigma ^2+i (x-x_0)}{\sqrt{2} \sigma }\right)$$ Trying to extract the real part of $K$ seems to be a small nightmare.

I have not been able to go beyond $$I=-\frac{1}{2} \sqrt{\frac{\pi }{2}} e^{-\frac{\sigma ^2}{2}} \sigma \left(e^{-i x_0} \text{erf}\left(\frac{x_0-i \sigma ^2-x}{\sqrt{2} \sigma }\right)+e^{i x_0} \text{erf}\left(\frac{x_0+i \sigma ^2-x}{\sqrt{2} \sigma }\right)\right)$$

$\endgroup$
  • $\begingroup$ @NitaRatnawaty thanks you (I think). $\endgroup$ – user228113 Jun 24 '15 at 10:35
  • $\begingroup$ You are welcome ! But I am not sure this helps much if you do not access the $\text{erf}$ function with complex arguments. Cheers :-) $\endgroup$ – Claude Leibovici Jun 24 '15 at 10:49
  • $\begingroup$ @G.Sassatelli. Thanks for the message ! $\endgroup$ – Claude Leibovici Jun 24 '15 at 10:50
0
$\begingroup$

Mathematica 8 says it is

$$ \int_{-L}^{L}\exp\!\left(-\frac{\left(x-x_{0}\right)^{2}}{2\sigma^{2}}\right)\cos\!\left(x\right)\mathrm{d}x = \frac{1}{2}\sqrt{\frac{\pi}{2}}s\exp\!\left(-\frac{1}{2}s^{2}-ix_{0}\right) \cdot $$ $$ \cdot \left[\exp\!\left(2ix_{0}\right)\mathrm{erf}\!\left(\frac{L+is^{2}+x_{0}}{\sqrt{2}s}\right)-i\left(\exp\left(2ix_0\right)\mathrm{erfi}\!\left(\frac{s^{2}+i(L-x_{0})}{\sqrt{2}s}\right)-\mathrm{erfi}\!\left(\frac{-iL+s^{2}+ix_{0}}{\sqrt{2}s}\right)+\mathrm{erfi}\!\left(\frac{iL+s^{2}+ix_{0}}{\sqrt{2}s}\right)\right)\right] $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.