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 .