To determine when $f(z) = \frac{z}{z^2 + 1}$ is analytic on an open set, we must check where the function is holomorphic. A function $f(z)$ is analytic (holomorphic) on an open set if it is differentiable at every point in that set.

Step 1: Identify singularities

The first step is to identify the points where the function may fail to be analytic. These are typically points where the function is undefined or not differentiable. Since the denominator is $z^2 + 1$, the function has potential singularities at the values of $z$ where:

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

Thus, the function has singularities at $z = i$ and $z = -i$.

Step 2: Analyticity on open sets

The function $f(z) = \frac{z}{z^2 + 1}$ is analytic everywhere in the complex plane except at $z = i$ and $z = -i$. Therefore, $f(z)$ is analytic on any open set that excludes these points. For instance, it is analytic on:

• $\mathbb{C} \setminus \{i, -i\}$, which is the complex plane excluding the points $i$ and $-i$,
• Any open set that does not include $z = i$ or $z = -i$.

Conclusion:

The function $f(z) = \frac{z}{z^2 + 1}$ is analytic on any open set that does not contain the points $z = i$ and $z = -i$, since these are the singularities where the function is undefined.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How can we classify the singularities of $f(z)$ at $z = i$ and $z = -i$?
2. How does the Laurent series expansion of $f(z)$ look around its singularities?
3. What are the conditions for a function to be meromorphic?
4. Can you find the residue of $f(z)$ at its singular points?
5. How would you apply the Cauchy-Riemann equations to check if $f(z)$ is analytic at specific points?

Tip: Always check where the denominator of a rational function becomes zero when analyzing the function's analytic properties.