- How to achieve convergence more quickly.
- 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 t_{0}, where t_{0}is a local maximum of Re h(t), such that Re h(t) decreases steeply on both sides of t_{0 }(we assume that |exp[z(h(t)]| is much greater than |g(t)|). - How to choose the path of steepest descent:
- Make the path pass through t
_{0}such that h'(t_{0})=0. - Let the small path segment near t
_{0}satisfy Im [h(t)]=Im [h(t_{0})].

Remark. This requirement will prevent oscillations [Guo, p.381, l.-15], make Re h(t) change most rapidly along the path near t_{0}[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].

- Make the path pass through t

- 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
- 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^{ }¢(b), and -
|ò(f-f
^{ }¢)| is arbitrarily small.

- ||f-f
- More precise formulas
- (Euler's constant) [Zyg, p.15, l.-2-l.-1] ® [Zyg, p.16, (8.9)].

- Links {1}.