Skip to content

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.


2. Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Any polynomial over complex field has at least one root.


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 \(\(f(a)=\dfrac{1}{2\pi}\int_0^{2\pi} f(a+re^{i\theta})d\theta\)\)


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