Define a branch of √
2 − z in the open connected set G = C \ {z ∈ R : z ≤ 2}.

To define a branch of the square root function \( f(z) = \sqrt{2 - z} \) in the open connected set \( G = \mathbb{C} \setminus \{z \in \mathbb{R} : z \leq 2\} \), we must ensure that \( f(z) \) is single-valued and analytic within the domain \( G \).

### Steps to Define the Branch:

1. **Domain Analysis**:  
   The set \( G \) excludes the real numbers less than or equal to 2, i.e., the negative real axis starting from 2 to \( -\infty \). The domain \( G \) is therefore the entire complex plane excluding the closed interval \( (-\infty, 2] \) on the real axis. This domain is simply connected, making it possible to define a branch of the square root function without ambiguity.

2. **Branch Cut**:  
   The standard branch cut for the square root function is usually taken along the negative real axis. However, since our function involves \( 2 - z \), the branch cut would naturally occur where \( 2 - z \) is non-positive, i.e., where \( z \geq 2 \) and \( z \) is real. However, since our domain excludes \( z \leq 2 \), the branch cut aligns with \( z = 2 \) and the positive real axis. This exclusion allows us to avoid crossing the branch cut, ensuring \( \sqrt{2 - z} \) is single-valued in \( G \).

3. **Defining the Branch**:  
   The square root function \( \sqrt{2 - z} \) can be defined on \( G \) by taking the principal branch of the square root function. The principal branch is defined with the argument \( \arg(2 - z) \) lying in the interval \( (-\pi, \pi] \). Specifically:
   \[
   f(z) = \sqrt{2 - z} = \exp\left(\frac{1}{2} \log(2 - z)\right)
   \]
   where \( \log(2 - z) \) is the principal branch of the logarithm, with \( \arg(2 - z) \) restricted to \( (-\pi, \pi] \).

4. **Properties**:  
   - \( f(z) \) is analytic in \( G \).
   - \( f(z) \) is single-valued in \( G \) since the branch cut prevents any ambiguity in the square root's argument.

### Final Definition:
The function \( f(z) = \sqrt{2 - z} \) on the domain \( G = \mathbb{C} \setminus \{z \in \mathbb{R} : z \leq 2\} \) is defined by the principal branch of the square root. This choice ensures that \( f(z) \) is analytic and single-valued in \( G \).

Would you like a more detailed explanation, or do you have any questions?

### Relative Questions:
1. What is the significance of choosing the principal branch for the square root function?
2. How does the exclusion of the interval \( (-\infty, 2] \) affect the analyticity of the function?
3. Can you explain the concept of a branch cut in the context of complex functions?
4. What would happen if we did not exclude the interval \( (-\infty, 2] \) from the domain?
5. How would the definition of the function change if we chose a different branch cut?

### Tip:
When dealing with multi-valued functions in complex analysis, carefully selecting the branch cut is crucial to ensuring the function remains single-valued and analytic within the desired domain.

Explore how to define a branch of the square root function sqrt(2 - z) in the open connected set G = C \ {z ∈ R : z ≤ 2}. Learn about branch cuts, analyticity, and the principal branch of the square root function 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