Cauchy Integral Theorem Calculator

What is Cauchy's Integral Theorem?

Cauchy's Integral Theorem states that if a function f(z) is holomorphic (analytic and differentiable) inside and on a simple closed contour C, then the integral of f(z) around C is zero:

The theorem has several important corollaries, including Cauchy's Integral Formula, which allows the reconstruction of a holomorphic function from its values on a contour.

How to Use This Calculator

To use the Cauchy Integral Theorem Calculator:

1. Enter the function f(z) that you want to evaluate
2. Define the closed contour C by specifying its parametric equations or vertices
3. Click "Calculate" to determine if the function satisfies the theorem conditions
4. View the result and interpretation

The calculator will verify if the function is holomorphic in the domain enclosed by the contour and compute the contour integral if applicable.

Formula Used

The calculator uses the following steps to evaluate Cauchy's Integral Theorem:

1. Check if the function f(z) is holomorphic in the domain enclosed by contour C
2. If holomorphic, compute the contour integral using the formula:

   ∮_C f(z) dz = 0

3. If not holomorphic, indicate that the theorem does not apply

Worked Example

Let's consider the function f(z) = z² and the unit circle contour C: z = e^(iθ) for θ ∈ [0, 2π].

Since z² is holomorphic everywhere in the complex plane, Cauchy's Integral Theorem applies:

This example demonstrates that for holomorphic functions, the contour integral around any simple closed contour is zero.

Applications of Cauchy's Theorem

Cauchy's Integral Theorem has numerous applications in complex analysis and mathematical physics, including:

• Deriving Cauchy's Integral Formula for reconstructing holomorphic functions
• Proving the Maximum Modulus Principle
• Analyzing residues in complex plane integrals
• Solving boundary value problems in potential theory