Here's an integral that is currently giving me grey hairs:
\int_0^{\infty} \frac{1}{x} \exp(i \frac{k}{x}(a-c \cos(\theta + wx))) dx
I've tried different approaches like contour integration around x=0 and replacing the exponential by its Taylor sum to have:
\int_0^{\infty}...
Hi! How do I approximate the integral
\begin{equation} \int_0^{\infty} dt \:e^{-iA(t-B)^2} \end{equation}
with A, B real, A > 0, and B=b \cos\theta where 0 \leq \theta < 2\pi?
I guess for B\ll 0 the lower limit may be extended to - \infty to yield a full complex gaussian integral, but what...
I'm trying to derive the x-space result for the Green's function for the Klein-Gordon equation, but my complex analysis skills seems to be insufficient. The result should be:
\begin{eqnarray}
G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int...