Treffer: Explicit Solutions for Transcendental Equations: Explicit solutions for transcendental equations

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.