Results Based on CIF¶

Cauchy Integral formula is prominent in complex analysis. It results in many properties of analytic functions.

1. Liouvilles Thorem¶

Liouvilles Thorem

Any bounded entire function is constant.

Proof:

2. Fundamental Theorem of Algebra¶

Fundamental Theorem of Algebra

Any polynomial over complex field has at least one root.

Proof:

3. Gauss Mean Value Theorem¶

Gauss Mean Value Theorem

If \(f\) is analytic inside and on a circular path centred at \(a\) with radius \(r\), then \(\(f(a)=\dfrac{1}{2\pi}\int_0^{2\pi} f(a+re^{i\theta})d\theta\)\)

Proof:

4. Maximum Modulus Theorems¶

Maximum Modulus Theorem

For an analytic function, the maximum value of modulus of the function occur on the boundary of the domain.

Proof: To prove this theorem we have to first prove the local maximum modulus theorem, which says the following

Local Maximum Modulus Theorem

For an analytic function defined on a domain \(\Omega\)

Last modified on: 2023-01-04 23:20:05