To determine if the function $f(z) = \frac{1}{1 - z^4}$ is analytic, we need to examine its behavior and identify any singularities.

1. Definition of Analytic Function:

A function is analytic (holomorphic) in a region if it is differentiable at every point in that region, meaning its derivative exists everywhere in the region and the function satisfies the Cauchy-Riemann equations.

2. Singularities:

First, let's check where $f(z)$ might not be differentiable, which occurs where the denominator equals zero:

$1 - z^4 = 0$

Solving this equation:

$z^4 = 1$

The solutions to this equation are the fourth roots of unity, which are:

$z = e^{2\pi i k / 4}, \quad k = 0, 1, 2, 3$

These correspond to the points:

$z = 1, \quad z = i, \quad z = -1, \quad z = -i$

3. Type of Singularities:

At these points, $f(z)$ has singularities, since the denominator becomes zero. To determine whether these singularities are poles or essential singularities, we examine the nature of these points.

We can factor the denominator as:

$1 - z^4 = (1 - z)(1 + z)(1 - iz)(1 + iz)$

Thus, the function $f(z) = \frac{1}{(1 - z)(1 + z)(1 - iz)(1 + iz)}$ has simple poles at the four points $z = 1, -1, i, -i$.

4. Analyticity:

For $f(z)$ to be analytic, it must be differentiable everywhere in a region. Since there are singularities at $z = 1, -1, i, -i$, the function is not analytic at these points. However, it is analytic in any region of the complex plane that does not include these points.

Conclusion:

$f(z) = \frac{1}{1 - z^4}$ is not analytic at $z = 1, -1, i, -i$, as these are singularities (poles). However, it is analytic in regions where these points are excluded.

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Would you like more details on how to identify the type of singularities or the Cauchy-Riemann equations for this function? Here are 5 related questions for further exploration:

1. How do you find the Laurent series around a singularity?
2. What are essential singularities and how do they differ from poles?
3. How do you use the Cauchy-Riemann equations to verify analyticity?
4. How can you determine the residue at a simple pole?
5. What is the concept of a branch cut in complex analysis?

Tip: Whenever checking for analyticity, always start by identifying the singularities of the function. If the function has poles or essential singularities, it cannot be analytic at those points.