I have the following problem. Consider the integral \begin{align} I(\tau, \xi)=\frac{\xi}{4\pi}\int_{\mathbb{R}}dz\frac{\exp(-i\tau\cosh{z}/\xi)}{\cosh^{3}{z}}, \quad \tau, \xi>0 . \end{align} This integral extremely looks like one of those know representation of Bessel/modified Bessel/Hankel functions or their combinations, however I was not able to find this particular one anywhere, so do it numerically. The only analytical insight I was intended to obtain is the asymptotic behaviour of the integral for $\tau\gg\xi$. For this sake we rewrite it as \begin{align} I(\tau, \xi)&=\frac{\xi}{4\pi}\int_{\mathbb{R}}dz\frac{\exp(-i\tau\cosh{z}/\xi)}{\cosh^{3}{z}}=\frac{\xi}{2\pi}\int_{0}^{\infty}dz\frac{\exp(-i\tau\cosh{z}/\xi)}{\cosh^{3}{z}}\\ &=\frac{\xi}{2\pi}\int_{1}^{\infty}dw\frac{\exp(-i\tau w/\xi)}{w^{3}\sqrt{w^{2}-1}}. \end{align} The integrand has a branch pole at $w=1$ on the integration contour. In order to analyse $\tau\gg\xi$ behaviour it is beneficial to analytically continue the square root into the lower half plane \begin{align} \sqrt{w^{2}-1}\rightarrow i\sqrt{-i(w+1)}\sqrt{-i(w-1)}. \end{align} Note that this indeed gives the disired result since $\sqrt{w^{2}-1}=i\sqrt{-i(w+1)}\sqrt{-i(w-1)}, \ w\in\mathbb{R}$ and the root has now the jump across the line $w=1-iy, \quad y\in\mathbb{R}$. In prticular, for $w=1-iy\pm\eta, \quad y\in\mathbb{R}$ we have \begin{align} i\sqrt{-i2-y}\sqrt{-y\mp i\eta}&=\pm i\sqrt{i2+y}\sqrt{y}\simeq \pm i(1+i)\sqrt{y}, \quad y\ll1,\\ (1-iy)^{3}&\simeq 1, \quad y\ll1. \end{align} It follows \begin{align} I(\tau, \xi)&\simeq-\underbrace{(-1)}_{\text{lower half}}\frac{\xi}{\pi}\int_{0}^{\infty}dy\frac{e^{-i(1-iy)v_{F}t/\xi}}{(1+i)\sqrt{y}}=\xi e^{-i(\tau/\xi+\pi/4)}\sqrt{\frac{\xi}{2\pi\tau}}, \quad \tau\gg\xi. \end{align} The thing is that this asymptotic does not agree with numerics unless it is multiplied by an extra factor of $1/2$. I.e. \begin{align} \frac{\xi}{2} e^{-i(\tau/\xi+\pi/4)}\sqrt{\frac{\xi}{2\pi\tau}}, \quad \tau\gg\xi, \end{align} approximates $I(\tau, \xi)$ perfectly for large $\tau/xi$....
Can anyone hint me where my mistake is? I mean factors of 1/2 from perfect agreement is clearly not an accident. Sorry in case the question is stupid in a way that it is based on my lack of elementary knowledge, or so, I'm just a physicist =)
Thanks in advance!