Asymptotic behavior of the integral $\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$ Which method can I use to study the asymptotic behavior as $\rho \to \infty$ of the integral for $q \geq 0$?
$$\int_{0}^{\infty} e^{-\rho \cosh(R)}\cosh(Rq)dR$$
I wish to study this behavior to understand more about modified Bessel functions and since I never studied in my life methods to study the behavior of functions, a detailed explanation on why the chosen method is the best to be applied will be very helpful, thank you.
 A: Assume that $\Re \rho>0$ and $q \in \mathbb C$ is fixed. Your integral is
$$
K_q (\rho ) = e^{ - \rho } \int_0^{ + \infty } {e^{ - \rho (\cosh (R) - 1)} \cosh (Rq)dR} .
$$
Now perform the change of variables $t = \frac{1}{2}(\cosh (R) - 1)$, to deduce
$$
K_q (\rho ) = e^{ - \rho } \int_0^{ + \infty } {e^{ - 2\rho t} t^{ - 1/2} \frac{{\cosh (q\cosh ^{ - 1} (1 + 2t))}}{{\sqrt {1 + t} }}dt} .
$$
Using $(15.4.13)$, we find
\begin{align*}
\frac{{\cosh (q\cosh ^{ - 1} (1 + 2t))}}{{\sqrt {1 + t} }} &= \frac{{(\sqrt {1 + t}  + \sqrt t )^{2q}  + (\sqrt {1 + t}  - \sqrt t )^{2q} }}{{2\sqrt {1 + t} }} \\ &= {}_2F_1 \!\left( {q + \tfrac{1}{2}, - q + \tfrac{1}{2};\tfrac{1}{2}; - t} \right),
\end{align*}
where ${}_2F_1$ is the Gauss hypergeometric function. Therefore,
$$
K_q (\rho ) = e^{ - \rho } \int_0^{ + \infty } {e^{ - 2\rho t} t^{ - 1/2} {}_2F_1\! \left( {q + \tfrac{1}{2}, - q + \tfrac{1}{2};\tfrac{1}{2}; - t} \right)dt} .
$$
Now you can apply Watson's lemma to deduce the familiar asymptotic expansion of the modified Bessel function $K_q (\rho )$ in the sector $|\arg \rho|\leq \frac{\pi}{2}-\delta<\frac{\pi}{2}$. By studying the properties of the hypergeometric function, you can actually show that this expansion is valid in the larger sector $|\arg \rho|\leq \frac{3\pi}{2}-\delta<\frac{3\pi}{2}$.
