Functions analytic in the whole plane are called entire . Polynomials are entire functions. Analytic functions have derivatives of any order which themselves turn out to be analytic functions. For analytic functions, this property is expressed by the Cauchy Integral Formula . Functions which are not entire have singularities in the finite plane.
Entire functions have a singularity at infinity.
The only ones that do not are constant . Polynomials have a pole at infinity. All entire functions with a pole of the same order behave in a similar manner. All polynomials of order n behave similarly to z n which has been exploited in somewhat different ways in the proofs by Cauchy and those taken from books by Birkhoff and MacLane and Courant and Robbins . Alexander Bogomolny 2469468 .