Complex Analysis/Global theory of holomorphic functions

Liouville's theorem Let be a Banach space, and let be an entire function. If there exists a natural number and two constants such that , then is a polynomial of degree less than or equal to . 1 First note that . Let and . Then Cauchy's integral formula and the triangle inequality for integrals together imply that for a certain . The latter expression may be computed explicitly; it equals , which tends to zero as . Hence, vanishes and is a polynomial of degree . identity theorem Let be a Banach space, let be open and connected, let and let be two holomorphic functions on such that the set has a cluster point . Then . Let be any point. Since holomorphic functions are analytic, the function posesses a power series expansion which converges on a sufficiently small neighbourhood of . Suppose first that is a cluster point of the set . Let be the least natural number such that . maximum principle argument principle Rouché's theorem Hurwitz's theorem Hartog's extension theorem Let , and let be holomorphic, where with . Then there exists a unique function such that . Since , we may pick the following subset of : , where is sufficiently small. Since the restriction of a holomorphic function is holomorphic, is holomorphic on . Moreover, Weierstraß preparation theorem Exercises Use Liouville's theorem to demonstrate that every non-constant polynomial has at least one root in (Hint: Consider the function ). In this exercise, we want to look at the simplest sufficient conditions for the possibility of extending a function given by a real power series to a function on the complex plane. Let be a power series with real coefficients which converges absolutely on an open neighbourhood of the origin of . Prove that may be extended to a function on an open neighbourhood of the origin of the complex plane. Let be a power series such that for all is real and positive. Suppose further that converges for all st. , where is a real number. Prove that may be extended to a holomorphic function on . Prove that the extensions considered in the first two sub-exercises are unique. Let be an entire function and let , and such that . Prove that is a polynomial of degree .