Functions and Analysis

The Solutions of Sinx=x in the Complex Domain

Authors: Bolok Li
Comments: 9 Pages.

We give a complete elementary treatment of the equation sin(x) = x over the complex numbers. We show that it has infinitely many solutions — exactly one, up to conjugation and sign, in each strip 2kπ < Re(z) < (4k+1)π/2, k ≥ 1 — among them only one real root, x = 0, and we supply an elementary proof of the uniqueness in each strip. Separating real and imaginary parts reduces the equation to two one-dimensional curve families whose intersections are precisely the solutions. We further derive explicit asymptotic expansions for the k-th solution: the real parts approach (4k+1)π/2 and the imaginary parts grow only logarithmically, b_k ~ ln(4πk). The leading term of the latter goes back to Hardy (1902); the higher-order terms, whose coefficients are polynomials in ln(2A_k) and which alternate in parity (odd powers for a_k, even powers for b_k), appear to be new, and are verified against roots recomputed at 40-digit precision. As a by-product of this verification we correct misprints in the classical tables of Hillman and Salzer (1943) and of Fettis (1976).
Category: Functions and Analysis