Cauchy's Formula

Cauchy's Integral Formula:

\[ f(z) = \sum_{n=0}^{\infty} a_n (z - c)^n \]

Complex Function f(z):

function

Center (c):

complex

Complex Variable (z):

variable

Coefficients (a_n):

sequence

Cauchy's Formula Result:

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1. What is Cauchy's Formula?

Cauchy's Integral Formula is a fundamental result in complex analysis that expresses an analytic function as a power series around a point. It provides the coefficients for the Taylor series expansion of analytic functions.

2. How Does the Formula Work?

The formula represents an analytic function as:

\[ f(z) = \sum_{n=0}^{\infty} a_n (z - c)^n \]

Where:

\( f(z) \) — Complex analytic function
\( a_n \) — Coefficients of the power series
\( z \) — Complex variable
\( c \) — Center of the series expansion
\( n \) — Non-negative integer index

Explanation: The formula shows that any analytic function can be represented as an infinite sum of powers around a center point c, with coefficients determined by the function's derivatives at that point.

3. Importance of Cauchy's Formula

Details: Cauchy's formula is crucial for understanding analytic functions, proving fundamental theorems in complex analysis, and providing methods for calculating complex integrals and derivatives.

4. Using the Calculator

Tips: Enter the complex function, center point, complex variable, and coefficients sequence. The calculator will display the power series representation of the function.

5. Frequently Asked Questions (FAQ)

Q1: What makes a function analytic?
A: A function is analytic at a point if it is complex differentiable in some neighborhood around that point and can be represented by a convergent power series.

Q2: How are the coefficients a_n determined?
A: The coefficients are given by \( a_n = \frac{f^{(n)}(c)}{n!} \), where \( f^{(n)}(c) \) is the nth derivative of f at point c.

Q3: What is the radius of convergence?
A: The radius of convergence is the distance from c to the nearest singularity of the function in the complex plane.

Q4: Can Cauchy's formula be used for real functions?
A: Yes, when restricted to real variables, it becomes the Taylor series expansion for real analytic functions.

Q5: What are the applications of Cauchy's formula?
A: Applications include solving differential equations, evaluating complex integrals, signal processing, and mathematical physics problems.