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The question asks whether \( z_0 \) is an isolated singularity of \( f(z) \) given that \( f(z) \) is analytic in the punctured disk \( 0 < |z - z_0| < r \), and whether this implies that \( f(z) \) has a Laurent series expansion in this domain.

In complex analysis, if \( f(z) \) is analytic in the punctured neighborhood around \( z_0 \), then \( z_0 \) is indeed an isolated singularity. Additionally, any isolated singularity allows for a Laurent series expansion of \( f(z) \) in \( 0 < |z - z_0| < r \).

Thus, the statement "has a Laurent series expansion" is **correct**. The answer should be:

- **B. ✅ (Correct)**

Would you like further details on Laurent series, types of singularities, or how to approach similar problems?

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Here are some related questions for further exploration:

1. What are the different types of isolated singularities?
2. How does a Laurent series differ from a Taylor series?
3. What does it mean if the Laurent series of \( f(z) \) around \( z_0 \) contains a principal part?
4. Can a singularity still be isolated if \( f(z) \) is not analytic in the punctured disk?
5. What are the conditions for a singularity to be classified as essential?

**Tip:** To determine the type of singularity, analyze the terms in the Laurent series expansion around that point.

z0 is an isolated singularity of f(z) if f(z) is analytic in the punctured disk 0 < |z - z0| < r. Determine if f(z) has a Laurent series expansion in this domain.

Explore whether the function f(z) has a Laurent series expansion around an isolated singularity z0 if it is analytic in the punctured disk 0 < |z - z0| < r. This problem covers essential concepts in complex analysis, including isolated singularities and the application of Laurent's theorem.

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