Precision in Differential Equations

  1. How to achieve convergence more quickly.
    1. Using the Euler formula [Guo, p.9, (6) & p.10, (10)].

  2. The method of steepest descent
    1. 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 |exp[z(h(t)]| is much greater than |g(t)|).
    2. How to choose the path of steepest descent:
      1. Make the path pass through t0 such that h'(t0)=0.
      2. 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].
      3. Choose the path of steepest descent [Guo, p.383, l.8-l.9].

  3. Given a sectionally smooth function f on [a,b]. Find a smooth function f such that
    1. ||f-f ||1 is arbitrarily small, where ||  ||1 is the strong norm,
    2. (Boundary conditions) f(a) = f (a); f(b) = f (b), and
    3. |(f-f )| is arbitrarily small.
    Method: Smoothing of corners [Sag, p.83, Theorem 2.10].

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

  5. Links {1}.