 Every entire function "f"("z") can be represented as a power series

that converges everywhere in the complex plane, hence uniformly on compact sets.The radius of convergence is infinite, which implies that

or

Any power series satisfying this criterion will represent an entire function.If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the complex conjugate of "z" will be the complex conjugate of the value at "z".