how to integrate from minus infinity to infinity. fourier in physics

Question

eamier · Answer

$$f(x)=e^\frac{ -x^2 }{ a^2 }$$
$$A(k)=\int\limits_{-\infty}^{\infty}f(x)\cos(kx)dx$$

eamier · Answer

i did until $$A(k)=\frac{ -a^2 }{ 2x }e^\frac{-x^2}{a^2}\cos(kx)-\int\limits_{-\infty}^{\infty}e^\frac{-x^2}{a^2}(-k)\sin(kx)dx$$

eamier · Answer

so should i substitute -infnity to infinity in the first set of term above or just ignore the limit

anonymous · Answer

You can't do it that way -- 
$$ \frac{-a^2}{2x} e^{-x^2/a^2} $$
is not correct.  
$$e^{-x^2/a^2} $$
doesn't have an elementary antiderivative.  I would recommend rewriting cosine in terms of exponentials, completing the square, and then using the formula for the Gaussian integral.  Otherwise, just use WolframAlpha or something.

eamier · Answer

i'll try