Power Series

Abstract

Power series in the complex variable $z$ are defined. Convergence is discussed, and the formula for the radius of convergence is derived. Example given include $e^z$ , $cos(z)$, and $sin(z)$. It is shown that these series can be differentiated term-by-term, a consequence of which is the Cauchy-Riemann equations which imply that in its circle of convergence a power series $f(z)$ has harmonic real and imaginary parts, a fact based on the complex variable boundary element method. A proof is given of the Weierstrass Approximation Theorem that every continuous function defined on a finite closed interval can be there uniformly approximated by polynomials