Taylor Series for eᶻ

Complex Analysis · Axiom Academy

Deriving the Maclaurin series of the complex exponential and finding its radius of convergence. Expand the complex exponential f(z) = e^z as a Taylor series centered at z = 0 (its Maclaurin series), and determine the radius of convergence of that series. Radius of convergence : the disk of convergence is the entire complex plane, so the series equals e^z at every point z . Nice work — you derived the Maclaurin series of the complex exponential and confirmed it converges everywhere. Self-derivative property: Since , every derivative equals e^z , so f^ (n) (0) = e^0 = 1 for all n . Simple coefficients: The Taylor coefficient , giving . Infinite radius: The ratio test gives for every z , so . An entire function: Because the power series converges on all of , e^z is analytic everywhere — an entire function. This single series underlies Euler's formula and the rest of complex analysis.

This is the written version of the interactive lesson above. See the full Complex Analysis course.