Contour Plots of Analytic Functions

There are two easy ways in Matlab to construct contour plots of analytic functions, i.e., lines of constant modulus and constant phase. One is to use the Matlab contour command for functions of two variables, another to solve the differential equations satisfied by the contour lines. This is illustrated here for the function f(z) = e n (z), where 25.1 $$ {e_{n}}(z) = 1 + z + \frac{{{z^{2}}}}{{2!}} + ... + \frac{{{z^{n}}}}{{n!}} $$ is the nth partial sum of the exponential series. The lines of constant modulus 1 of e n are of interest in the numerical solution of ordinary differential equations, where they delineate regions of absolute stability for the Taylor expansion method of order n and also for any n-stage explicit Runge-Kutta method of order n, 1 ≤ n ≤ 4 (cf. [4, §9.3.2]).