91Ó°ÊÓ

Use Watson's Lemma to obtain an infinite asymptotic expansion of $$ I(k)=\int_{0}^{\pi} e^{-k t} t^{-\frac{1}{3}} \cos t d t $$ as \(k \rightarrow \infty\). Note that $$ \cos t=\sum_{n=0}^{\infty}(-1)^{n} \frac{t^{2 n}}{(2 n) !} \text { for }-\infty

Use Watson's lemma to find an infinite asymptotic expansion of $$ I(k)=\int_{0}^{9} \frac{e^{-k t}}{1+t^{4}} d t $$ as \(k \rightarrow \infty\).

Use Laplace's method to determine the leading behavior (first term) of $$ I(k)=\int_{-\frac{1}{2}}^{\frac{1}{2}} e^{-k \sin ^{4} t} d t $$ as \(k \rightarrow \infty\).

Parseval's formula for Mellin transforms (see Problem 5c): Consider the integral $$ I=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} H(s) F(1-s) d s $$ where \(H(s), F(s)\) are the Mellin transforms associated with \(h(t), f(t)\), respectively, and assume \(H(s), F(1-s)\) are analytic in some common vertical strip: \(\alpha