Math 6420 • PDE

Consider integral
\[
  I(s)=\int_C \frac{e^{s(z^2-1)}}{z-1}\,dz
\]
with large parameter \(s\to+\infty\); \(C\) is the vertical line \(\operatorname{Im}z=3\),
from \(z=3-i\infty\) to \(z=3+i\infty\).
\begin{enumerate}
\item[(a)] What is the saddle point \(z_0\)?

What is the path \(C_0\) of steepest descent from \(z_0\)?

What is the path of steepest ascent from \(z_0\)?

\item[(b)] Find the three-term asymtotic expansion of the integral.
\end{enumerate}