Proving exp(z) is Entire

Complex Analysis · Axiom Academy

A rigorous demonstration that the exponential is analytic everywhere, using the Cauchy–Riemann equations. Prove that f(z) = e^z is an entire function — that is, it is analytic at every point of the complex plane . We'll do it by splitting e^z into real and imaginary parts and checking the Cauchy–Riemann equations. Nice work — you proved a cornerstone result of complex analysis by hand. The Cauchy–Riemann test plus continuity of the partials does all the heavy lifting. The recipe: Write f = u + iv , compute the four partials, check u_x = v_y and u_y = -v_x , and confirm the partials are continuous. Why e^z passes everywhere: and hold at every point of — no excluded points — so e^z is entire. Continuity matters: all four partials are products of e^x and a trig function, hence continuous on all of . C–R alone isn't enough; the theorem needs continuous partials. The derivative drops out: , recovering (e^z)' = e^z . e^z can be defined by . The ratio test gives for every fixed z , so the radius of convergence is . A power series is analytic inside its disk of convergence and may be differentiated term by term. With , e^z is analytic on all of — entire — and term-by-term differentiation again yields (e^z)' = e^z . This route is cleaner, while the Cauchy–Riemann proof shows exactly how the real and imaginary parts cooperate.

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