Cauchy's Integral Formula

Cauchy's Integral Formula is an important concept in complex analysis, and it has significant applications in calculus and mathematical physics.

Definition

The Cauchy's Integral Formula states that if $f(z)$ is a continuous function on a simple closed curve $C$, and $z_0$ is a point inside $C$, then for any point $z$ inside $C$, we have the following formula:

Where $C$ is the simple closed curve, $\zeta$ is the dummy variable of integration, and $f(\zeta)$ is the function.

Significance

This formula provides a powerful method for evaluating complex integrals and is used in various applications such as evaluating improper integrals, calculating residues, solving differential equations, and more.

Example

Consider the function $f(z) = \frac{1}{z}$ and a simple closed curve $C$ where $|z| = 2$. We can use Cauchy's Integral Formula to evaluate the following integral:

By applying the formula, we can simplify the integral and calculate the value using the provided formula.

Conclusion

Cauchy's Integral Formula is a fundamental result in complex analysis that plays a crucial role in calculus and mathematical physics. Understanding this formula allows us to solve complex integrals and analyze complex functions efficiently.