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Complex function theory and numerical analysis. (English) Zbl 1100.65020
Summary: What we deal with in computation in physics, for example, is usually a function expressed in terms of a single formula like an algebraic function or an elementary transcendental function, etc. In some cases it involves the symbol of differentiation or that of integration, or it is given implicitly as a solution of some differential equation whose coefficients consist of such functions. In any case such a function is an analytic function handled in the complex function theory.
Actually I dare to say that more than 90 percent of practical applications of mathematical analysis deal with analytic functions. Nevertheless, in conventional textbooks on numerical analysis almost all functional algorithms such as interpolation, numerical integration, and numerical differentiation are dealt with in the framework of only the elementary calculus or, in other words, techniques in the theory of real functions.
This seems strange to me. We have successfully shown that methods based on the complex function theory are quite efficient in a number of problems of numerical computation, in particular in numerical integration. In what follows, we show other examples in which methods based on the complex function theory are useful and also point out some flaws of the conventional methods.

MSC:
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
65D05 Numerical interpolation
30B10 Power series (including lacunary series) in one complex variable
30B40 Analytic continuation of functions of one complex variable
30E10 Approximation in the complex plane
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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References:
[1] H. Takahasi & M. Mori: Error estimation in the numerical integration of analytic functions, Report of the Com- puter Centre, Univ. of Tokyo 3 (1970), 41-108; Estimation of errors in the numerical quadrature of analytic functions, Applicable Anal- ysis 1 (1971), 201-229; Quadrature formulas obtained by variable transformation, Numerische Mathematik 21 (1973), 206-219; Double exponential formulas for numerical integration, Publ. of RIMS, Kyoto Univ. 9 (1974), 721-741.
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