- The Euler-Maclaurin Summation Formula and Bernoulli Polynomials

Read http://www.serc.iisc.ernet.in/~amohanty/SE288/emsf.pdf - Successive approximations for a second-order ODE

Goal: [Wat1, p.195, l.-7]

Design: [Wat1, p.195, l.6]

Principle based on: successive approximations

key idea: (d/dz)ò_{[b, z]}(z-z) f(z)dz = -ò_{[b, z]}f(z)dz. - The reason for distinguishing regular singular points from irregular singular points
[Wat1, p.198, l.15-l.19]

If we want to appreciate the distinction between regular and irregular singular points, we must create an*example*, say

2x^{3}y" + (1+x)y'+3xy = 0, whose indicial equation gives only*one*root for a. - Integral transforms [Guo, §2.12]

Goal: Solve ODE's by integrals [Guo, p.79, (9)].

Design: [Guo, p.79, l.11-p.80, l.10]

Principle based on: [Guo, p.79, (6)]

Key idea: [Guo, p.80, l.14-l.19]

Examples: [Guo, p.80, l.-9-l.-1] - A contour design for the case given in [Guo, p.82, (ii)]

Goal: Solve ODE's using the Laplace transform. More specifically, the contour must satisfy the condition given in [Guo, p.82, (7)].

Design: [Guo, p.83, Fig. 3]

Principle based on: a_{1}is a branch point of the Riemann surface for w = log (z-a_{1}). If a point P goes along the contour circling a_{1}counterclockwise once and then clockwise once, P will remain on the same sheet of the Riemann surface. Consequently, the starting point and the ending point of the contour given in [Guo, p.83, Fig. 3] will have the same value for log (z-a_{1}).

Key ideas: It is unnecessary to consider the Riemann surface for w = (z-a_{1})^{l1}(z-a_{2})^{l2}[Guo, p.83, l.2-l.3]. All we have to do is paste the contour given in [Guo, p.83, Fig. 3] on the Riemann surface for w = log (z-a_{1}). After considering (z-a_{1})^{l1}, we paste the contour on the Riemann surface for w = log (z-a_{2}).

Remark. Instead of using the hint given in [Guo, p.83, l.5], I use [Ru2, p.27 Theorem 1.34] to justify the differentiation under the integral sign given in [Guo, p.83, (12)]. - Differentiability of angle functions [O'N, p.50, Exercise 12]
- The possible radical change of a curve at the point whose curvature is zero [O'N, p.76, Exercise 19]