In the expression

$f(z) = \frac{\sin(\pi z)}{z \left( z^2 + 1 \right)^2},$

the variable $z$ cannot be equal to 0 for the following reasons:

1. Singularity at $z = 0$: The term $\frac{1}{z}$ in the denominator will become undefined when $z = 0$. Division by zero is undefined in mathematics, making the expression singular at this point.

   Specifically, the denominator $z \left( z^2 + 1 \right)^2$ becomes zero when $z = 0$ because the $z$-term directly multiplies the whole denominator, causing the whole denominator to equal zero.

2. Removable Singularity: Even though $\sin(\pi z)$ is finite and continuous at $z = 0$ (since $\sin(0) = 0$), this does not resolve the issue of dividing by zero. The singularity is not automatically removable unless the entire numerator approaches zero fast enough to cancel the $z$ in the denominator.

Therefore, $z = 0$ is a singularity where the function is not defined, so we must restrict $z \neq 0$.

Would you like further details or have any questions?

Related Questions:

1. What is the type of singularity at $z = 0$ for this function?
2. How do we classify singularities in complex functions?
3. What happens to the function's behavior as $z \to \infty$?
4. How can we check if a singularity is removable or not?
5. What would be the limit of $f(z)$ as $z \to 0$ if we could remove the singularity?

Tip:

Always check for terms in the denominator when determining where a function is undefined.