Using the Euler formula [Guo, p.9, (6) & p.10, (10)].
The method of steepest descent
Key idea: We deform the path of the integral [Guo, p.381, (1)] into the path of steepest
descent [Guo, p.383, l.7]. By Cauchy's theorem, the value of the integral will
remain the same. Our goal is to find a small path segment on which the
value of the integral is sufficient to approximate the original integral with great accuracy. This accuracy
is possible when the
contribution of the integrand outside the small path segment is insignificant.
In order to achieve this goal, we would like to select a small path segment passing
through t0, where t0
is a local
maximum of Re h(t), such that Re h(t) decreases steeply on both
sides of t0 (we assume that
is much greater
How to choose the path of steepest descent:
Make the path pass through t0 such that
Let the small path segment near t0 satisfy
Im [h(t)]=Im [h(t0)].
Remark. This requirement will prevent oscillations [Guo, p.381, l.-15],
make Re h(t) change most rapidly along the path
near t0 [Guo, p.382, l.4], and yield two
steepest paths [Guo,
p.382, l.-10-p.383, l.4].
Choose the path of steepest descent [Guo, p.383, l.8-l.9].
Given a sectionally smooth function f on [a,b]. Find a smooth function f¢ such that
||f-f¢||1 is arbitrarily small,
where || ||1
is the strong norm,
(Boundary conditions) f(a) = f¢(a); f(b) = f
|ò(f-f¢)| is arbitrarily small.
Method: Smoothing of corners [Sag, p.83, Theorem