# Complex Analysis/Global theory of holomorphic functions

**Theorem (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 .

**Proof 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 .

**Theorem (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 .

**Proof:** 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 .

**Theorem (maximum principle)**:

**Theorem (argument principle)**:

**Theorem (Rouché's theorem)**:

**Theorem (Hurwitz's theorem)**:

**Theorem (Hartog's extension theorem)**:

Let , and let be holomorphic, where with . Then there exists a unique function such that

- .

**Proof:** 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,

**Theorem (Weierstraß preparation theorem)**:

## Exercises Edit

- 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 .