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Contour Plots of Analytic Functions

by W. Gautschi and J. Waldvogel

(Report number 1996-15)

Abstract
This is a tutorial on generating contour lines of an analytic function $f(z)$. The emphasis is on using mathematical software (MATLAB, to a lesser extent MAPLE) for implementing the algorithms, and efficient programs together with explanations are presented. Two different approaches are suggested: (1) generating level lines as contours of, e.g., constant modulus or constant phase of the function $f(z)$, (2) setting up and numerically integrating an appropriate differential equation for the contour under consideration. Both methods are demonstrated by means of the $n$th partial sum $f(z)=e_n(z)$ of the exponential series. The line of constant modulus satisfying $|e_n(z)|=1$ has a practical significance: it delineates the region of absolute stability for an explicit Taylor integrator of order $n$.

Keywords:

@Techreport{GW96_198,
  author = {W. Gautschi and J. Waldvogel},
  title = {Contour Plots of Analytic Functions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1996-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1996/1996-15.pdf },
  year = {1996}
}

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