State Watson's lemma, describing the asymptotic behaviour of the integral

$$I(\lambda)=\int_{0}^{A} e^{-\lambda t} f(t) d t, \quad A>0$$

as $\lambda \rightarrow \infty$, given that $f(t)$ has the asymptotic expansion

$$f(t) \sim t^{\alpha} \sum_{n=0}^{\infty} a_{n} t^{n \beta}$$

as $t \rightarrow 0_{+}$, where $\beta>0$ and $\alpha>-1$.

Give an account of Laplace's method for finding asymptotic expansions of integrals of the form

$$J(z)=\int_{-\infty}^{\infty} e^{-z p(t)} q(t) d t$$

for large real $z$, where $p(t)$ is real for real $t$.

Deduce the following asymptotic expansion of the contour integral

$$\int_{-\infty-i \pi}^{\infty+i \pi} \exp (z \cosh t) d t=2^{1 / 2} i e^{z} \Gamma\left(\frac{1}{2}\right)\left[z^{-1 / 2}+\frac{1}{8} z^{-3 / 2}+O\left(z^{-5 / 2}\right)\right]$$

as $z \rightarrow \infty$.