To determine the values of $z$ for which the function

$f(z) = \frac{z^2 - 4}{z^2 + 1}$

is analytic, we need to identify where this function is not analytic. A function is analytic at points where it is differentiable, meaning it must be holomorphic (complex differentiable) in a region.

For this function to be analytic, it must be defined and differentiable. The only issue here comes from points where the denominator $z^2 + 1$ is zero, since division by zero makes the function undefined at those points.

Step 1: Solve for when the denominator equals zero.

Set the denominator equal to zero and solve for $z$:

$z^2 + 1 = 0$ $z^2 = -1$ $z = \pm i$

So, the function $f(z)$ is undefined at $z = i$ and $z = -i$.

Step 2: Analyticity of $f(z)$

The function $f(z)$ is analytic everywhere except at $z = i$ and $z = -i$, where the function has singularities (poles).

Thus, the function is analytic for all $z \in \mathbb{C}$ except $z = \pm i$.

Conclusion:

The function $f(z) = \frac{z^2 - 4}{z^2 + 1}$ is analytic for all complex values of $z$ except at $z = i$ and $z = -i$, i.e., $z \neq \pm i$.

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Would you like further clarification on the concept of analyticity or any other details about complex functions?

Here are some related questions to expand your understanding:

1. What is the definition of a singularity in complex analysis?
2. How do you classify the types of singularities a function might have?
3. Can a function be analytic on a domain with singularities? If so, under what conditions?
4. What is a pole, and how is it different from an essential singularity?
5. How do you compute the residue of a function at a pole?

Tip: When dealing with functions involving complex fractions, always look for values that make the denominator zero, as these points are where the function might not be analytic.