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.