To determine the type of singularity at $z = 0$ for the given function

$f(z) = \left(e^{z^3} - 1 - z^3 \right) \cdot \cot(2z^3),$

we analyze the behavior of the function around $z = 0$:

Step-by-step Analysis:

1. Expand $e^{z^3}$ in a Taylor series around $z = 0$: $e^{z^3} = 1 + z^3 + \frac{(z^3)^2}{2!} + \frac{(z^3)^3}{3!} + \cdots$ Therefore: $e^{z^3} - 1 - z^3 = \frac{(z^3)^2}{2!} + \frac{(z^3)^3}{3!} + \cdots = \frac{z^6}{2} + \frac{z^9}{6} + \cdots.$

2. Simplify the numerator of $f(z)$: The numerator of $f(z)$ becomes: $\left(e^{z^3} - 1 - z^3\right) = \frac{z^6}{2} + \frac{z^9}{6} + \cdots.$

3. Expand $\cot(2z^3)$: Recall the series expansion of $\cot(x)$: $\cot(x) = \frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} + \cdots.$ Substituting $x = 2z^3$: $\cot(2z^3) = \frac{1}{2z^3} - \frac{2z^3}{3} - \frac{(2z^3)^3}{45} + \cdots.$

4. Behavior of $f(z)$ near $z = 0$: Combining the two terms, the function becomes: $f(z) = \left(\frac{z^6}{2} + \frac{z^9}{6} + \cdots \right) \cdot \left(\frac{1}{2z^3} - \frac{2z^3}{3} - \frac{(2z^3)^3}{45} + \cdots \right).$ Multiply the terms:

   • The leading term in the product is: $\frac{z^6}{2} \cdot \frac{1}{2z^3} = \frac{z^3}{4}.$
   • Higher-order terms are of the form $z^6, z^9,$ etc.

   Hence, near $z = 0$, $f(z) \sim \frac{z^3}{4}$, meaning $z = 0$ is a removable singularity because the function is analytic there after division by $z^3$.

Conclusion:

The singularity at $z = 0$ for the given function is a removable singularity.

Would you like further clarification or additional examples?

Related Questions:

1. What are the different types of singularities in complex analysis?
2. How can Taylor expansions help analyze singularities?
3. How does the behavior of $\cot(z)$ near $z = 0$ affect singularities?
4. Could this function have other singularities besides $z = 0$?
5. How do Laurent series differ from Taylor series in classifying singularities?

Tip:

When analyzing singularities, always focus on the leading-order term of the Laurent series to determine the type of singularity.