Approximations in Differential Equations

  1. In order to use Cauchy's theorem [Ru2, p.223, Theorem 10.14] in complex analysis, we do not like singularities. Using the concept of approximation, we may avoid singularities and can still use Cauchy's theorem.
    Example. [Guo, p.264, Fig. 18] is designed to illustrate the statement given in [Wat1, p.315, l.10-l.12].
    Even though eiq and e-iq are singularities of the integrand of the integral given in [Wat1, p.315, l.5], we may avoid singularities by using the contour made by x = cos q and  h = reiq (Re h ³ cos q; 0<r<1) instead of the contour in [Guo, p.264, Fig. 18]. Then let r®1-. [Wat1, p.315, l.-3-l.-1] provides another method of approximation.