Explicit solutions for transcendental equations. (English) Zbl 1094.30008

Luck, Rogelio; Stevens, James W.

SIAM Rev. 44, No. 2, 227-233 (2002).

The approach presented here determines the roots of a transcendental function by locating the singularities of the reciprocal of the function. If an analytic function, \(f(z)\), contains a single simple pole at \(z_0\) somewhere inside \(C\), then the singularity can be removed by multiplying \(f(z)\) by \((z- z_0)\). Cauchy’s theorem implies that the path integral of the new function around \(C\) must be zero: \(\oint_C (z-z_0)f(z)\,dz = 0.\) Solving this equation for \(z_0\) yields an explicit expression for the singularity of \(f(z)\): \[ z_0=\frac{\oint_C zf(z)\,dz}{\oint_C f(z)\,dz}.\tag{1} \] A root finding problem may be recast as a singularity at the root, and (1) yields the desired root. The expression (1) can be evaluated over any closed path and with any technique, analytical or numerical, that is convenient. One strategy for evaluation of (1) uses a circle in the complex plane that circumscribes the root and requires only standard complex fast Fourier transform. For an illustration, the transcendental equations \(z\tan(z)=B\), \((5-z)e^z=5\) and \(ze^{z^2}\text{erf}(z)=S/\sqrt{\pi}\) are considered. The procedure is conceptually simple and can easily be implemented with the tools commonly available in commercial mathematical packages.

Reviewer: Eleonora A. Storozhenko (Odessa)

MSC:

30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30E20	Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
65H05	Numerical computation of solutions to single equations

Keywords:

analytic functions; transcendental equations; Cauchy integral theorem; pole-zero cancellation; complex fast Fourier transform