Entire Functions

Complex Analysis · Axiom Academy

The functions that are complex-differentiable everywhere — no poles, no branch cuts, no exceptions. 1. Analytic Everywhere, No Exceptions A function is entire if it is analytic at every single point of — equivalently, if the complex derivative f'(z) exists for all z . There are no singular points to avoid. Watch a probe travel the plane. For the entire function e^z (left), the derivative exists at every point it visits. For 1/z (right) the same probe runs into a pole at the origin, where the function blows up — so 1/z is not entire. 2. The Exponential: The Archetypal Entire Function Polynomials, e^z , , , the hyperbolic functions, and any power series with infinite radius of convergence are all entire — and entire-composed-with-entire stays entire. The exponential e^z shows off every trait at once. (e^z)' = e^z exists for all z . e^z omits exactly one value, 0 . Each point z=x+iy below is colored by where e^z sends it: the hue is the argument , the brightness is the modulus |e^z|=e^x . The picture is smooth across the whole plane — no holes, no tears, no seams — exactly because e^z is entire. Which functions are — and aren't — entire Always entire; the derivative is again a polynomial. . Unbounded on , unlike the real case. Poles where the denominator vanishes — meromorphic, not entire. Branch point at z=0 ; need a branch cut to be single-valued. Poles wherever — infinitely many singularities. 3. Liouville: Bounded Entire Constant

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