Generalization of Cauchy’s Integral Formula
Generalization of Cauchy’s Integral Formula allows for the computation of higher-order derivatives of an analytic function using contour integrals. This is particularly useful when dealing with complex functions in mathematical analysis and applications in physics and engineering.
Original Cauchy’s Integral Formula deals specifically with the first derivative of an analytic function, while the generalized form extends this concept to higher-order derivatives. This means that with the generalization, we can compute not only the first derivative but also the second, third, and so on, derivatives of the function at a given point within a contour.
A generalization of Cauchy’s Integral Formula is known as the Cauchy Integral Formula for Derivatives, which expresses higher-order derivatives of an analytic function in terms of contour integrals. It is given by:
[Tex]f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z – z_0)^{n+1}} \, dz [/Tex]
Cauchy Theorem
the Cauchy-GoursatCauchy’s Theorem states that if a function is analytic within a closed contour and its interior, then the integral of that function around the contour is zero. This fundamental result is central to complex analysis, providing insights into the behavior of analytic functions in complex domains.
In this article, we will learn about Cauchy’s Integral Theorem, Cauchy’s Integral Formula, It’s applications, Cauchy’s Residue Theorem and the Cauchy-Goursat Theorem.
Table of Content
- What is Cauchy’s Integral Theorem?
- Cauchy’s Integral Formula
- Applications of Cauchy Theorem
- Cauchy’s Residue Theorem
- Extension of Cauchy’s Theorem