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 (Fundamental Theorem for Complex Integrals):

Cauchy's Integral Theorem states that if \( f(z) \) is holomorphic (analytic) on and inside a simple closed contour \( C \), then the integral of \( f(z) \) along \( C \) is zero. In other words, if \( f(z) \) is analytic everywhere inside and on a simple closed curve \( C \), then:

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

#### Conditions for the theorem to hold:
1. The function \( f(z) \) must be analytic (holomorphic) inside and on the contour \( C \).
2. The contour \( C \) must be closed and simple, meaning it doesn't intersect itself and encloses a well-defined region.

#### Justification for the function \( f(z) = z^3 + i z^2 - 5z + 2i \) over the circle \( |z| = 1 \):

Let's check if Cauchy's Integral Theorem applies to the function \( f(z) = z^3 + i z^2 - 5z + 2i \), where \( C \) is the circle \( |z| = 1 \).

1. **Holomorphy**: The function \( f(z) = z^3 + i z^2 - 5z + 2i \) is a polynomial in \( z \), and polynomials are entire functions. An entire function is holomorphic everywhere in the complex plane, including inside and on the contour \( C \).
   
   Since \( f(z) \) is analytic on and inside the contour \( |z| = 1 \), it satisfies the conditions for Cauchy's Integral Theorem.

2. **Conclusion based on the theorem**: Since \( f(z) \) is holomorphic inside and on the contour \( C \), Cauchy's Integral Theorem applies. Therefore, the integral of \( f(z) \) over the circle \( |z| = 1 \) must be zero:

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

This result holds because \( f(z) \) is analytic everywhere in the complex plane, satisfying all the conditions for the application of Cauchy's theorem.

### Final Answer:

For the function \( f(z) = z^3 + i z^2 - 5z + 2i \) and the contour \( C \) being the circle \( |z| = 1 \), the integral of \( f(z) \) around \( C \) is zero based on Cauchy's Integral Theorem:

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

---

Would you like further details or clarification on any part of this explanation?

Here are 5 follow-up questions to deepen your understanding:
1. How do you determine if a function is holomorphic in general?
2. Can Cauchy's Integral Theorem apply if a function has a singularity inside the contour \( C \)?
3. What are some consequences of Cauchy's Integral Theorem in complex analysis?
4. How would you compute the integral if \( f(z) \) had singularities inside the contour \( C \)?
5. What is the difference between Cauchy's Integral Theorem and Cauchy's Integral Formula?

**Tip**: When working with complex integrals, always check if the function is analytic within the region enclosed by the contour—this is crucial for applying Cauchy's theorems.

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

Understand Cauchy's Integral Theorem and learn how to apply it to evaluate the complex integral of f(z) = z^3 + i(z)^2 - 5z + 2i over the contour |z| = 1. This comprehensive explanation covers key concepts in complex analysis and provides a step-by-step justification for the theorem. Suitable for undergraduate mathematics 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