How to get the short time asymptotics of this integral?

Question

The integral is like this:

$$ \int_0^\infty \mathrm{d} x \frac{\cos[2t\cosh(\frac{\pi x}{2})]}{1+x^2} $$

The short time asymptotics is like this (some constant maybe missing):

$$ \sim \frac{1}{\ln t} $$

I don't know how to get this...

Edit: what's the meaning of short time?

The source I am reading says:

At short times the integral rises or falls sharply as $\sim \frac{1}{\ln t}$

My guess that the meaning of short time is the time smaller than the first maximum or minimum of the integral.

Answer 1

here is an idea: i will look at the asymptotics of the simpler $$I= \int_0^\infty \frac{\cos(te^x)}{1+x^2}\, dx \text{ for small } t. $$

observe that the graph of $y = \frac{\cos(te^x)}{1+x^2}$ oscillates between the envelopes $y = \pm \frac 1{1+x^2}$ starts at $(0,1)$ and the first $x$ -intercept is $\ln(\pi/2) - \ln(t).$ there after the intercepts are spaced by $\ln 2, \ln 3. \cdots.$

therefore $$I= \int_0^\infty \frac{\cos(te^x)}{1+x^2}\, dx=\int_0^{\ln(\pi/(2t)}\frac{\cos(te^x)}{1+x^2}\, dx +\cdots = \frac12 \times 1 \times \ln\left(\frac{\pi}{2t}\right) + \cdots$$ the estimate being the area of the triangle with base $ \ln\left(\frac{\pi}{2t}\right)$ and height $1.$

Answer 2

For $x$ real we have

$$ \left| \frac{\cos[2t\cosh(\frac{\pi x}{2})]}{1+x^2} \right| \leq \frac{1}{1+x^2}, $$

we know that

$$ \frac{1}{1+x^2} \in L^1([0,\infty)), $$

and for fixed $x \geq 0$ we calculate

$$ \lim_{t \to 0^+} \frac{\cos[2t\cosh(\frac{\pi x}{2})]}{1+x^2} = \frac{1}{1+x^2}. $$

Therefore we may conclude that

$$ \lim_{t \to 0^+} \int_0^\infty dx \frac{\cos[2t\cosh(\frac{\pi x}{2})]}{1+x^2} = \int_0^\infty \frac{dx}{1+x^2} = \frac{\pi}{2}. $$

by the Dominated Convergence Theorem.