Describe cauchy's fundamental theorem for evaluating complex intregal and justify the theorem for the function :

f(z) = z^3 + i(z)^2 -5z + 2i if C is the circle  |z| = 1 described in the positive sense

**Cauchy's Integral Theorem** (also known as Cauchy's Fundamental Theorem) is a cornerstone in complex analysis. It states that if \( f(z) \) is a holomorphic (analytic) function on and inside a simple closed curve \( C \), then the contour integral of \( f(z) \) around \( C \) is zero:

\[
\int_C f(z) \, dz = 0
\]

### Conditions for Cauchy's Theorem:
1. \( f(z) \) must be holomorphic (analytic) inside and on the curve \( C \).
2. \( C \) is a positively oriented, simple closed contour (meaning it encloses a region, typically traversed counterclockwise).

### Justification for the Function \( f(z) = z^3 + i z^2 - 5z + 2i \):

Given the function \( f(z) = z^3 + iz^2 - 5z + 2i \) and the contour \( C \) as the circle \( |z| = 1 \), we need to verify whether \( f(z) \) satisfies the conditions of Cauchy's Theorem:

1. **Analyticity**:  
   \( f(z) \) is a polynomial in \( z \), which is holomorphic (analytic) everywhere in the complex plane. Specifically, it is analytic inside and on the contour \( C \) (the circle \( |z| = 1 \)). There are no singularities inside or on \( C \).

2. **Closed Contour**:  


Describe Cauchy's fundamental theorem for evaluating complex integral and justify the theorem for the function: f(z) = z^3 + i(z)^2 - 5z + 2i if C is the circle |z| = 1 described in the positive sense.

Explore Cauchy's Fundamental Theorem in complex analysis and its application to the integral of the function f(z) = z^3 + iz^2 - 5z + 2i over the contour C, the unit circle |z| = 1. Learn about holomorphic functions and contour integration, suitable for undergraduate and graduate-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation