Give an example of a function f(z) that is analytic in a region, bounded throughout the complex plane
but is not a constant function. Does this contradict the Liouville’s theorem? Explain.

To find a function \( f(z) \) that is analytic in a region, bounded throughout the complex plane, but is not a constant function, we can use the following example:

\[ f(z) = \frac{1}{1+z^2} \]

Let's analyze this function:

1. **Analytic in a Region**:
   - The function \( f(z) = \frac{1}{1+z^2} \) is analytic wherever the denominator is not zero.
   - The denominator \( 1+z^2 = 0 \) when \( z = \pm i \) (where \( i \) is the imaginary unit).
   - Therefore, \( f(z

Explore an example of an analytic function f(z) = 1/(1+z^2) that is bounded in the complex plane but not constant. Learn how this function relates to Liouville's Theorem in complex analysis. Suitable for advanced undergraduate students and those interested in complex analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation