Results Based on CIF

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

Liouvilles Thorem

Liouvilles Thorem: Any bounded entire function is constant.

Proof:

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra: Any polynomial over complex field has at least one root.

Proof:

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:

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 $\Omega$