Complex Analysis/Global theory of holomorphic functions

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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]

  1. Use Liouville's theorem to demonstrate that every non-constant polynomial has at least one root in (Hint: Consider the function ).
  2. 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.
    1. 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.
    2. 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 .
    3. Prove that the extensions considered in the first two sub-exercises are unique.
  3. Let be an entire function and let , and such that . Prove that is a polynomial of degree .