Question

Say whether the indicated point is regular, an essential singularity, or a pole, and if a pole of what order it is. (a) \(\frac{e^{z}-1}{z^{2}+4}, \quad z=2 i\) (b) \(\tan ^{2} z, \quad z=\pi / 2\) (c) \(\frac{1-\cos z}{z^{4}}, \quad z=0\) (d) \(\cos \left(\frac{\pi}{z-\pi}\right), \quad z=\pi\)

Short answer

(a) pole of order 1; (b) pole of order 2; (c) pole of order 2; (d) essential singularity

Step-by-step solution

Analyze the function at the given point

For each function, examine the type of singularity at the given point and determine if it is a regular point, essential singularity, or pole.

(a) \(\frac{e^{z}-1}{z^{2}+4}, \ z=2i\)

Evaluate the denominator at the point \(2i\). Since \(z^2 + 4 = 0\) or \(4i^2 + 4 = -4 + 4 = 0\), the denominator is zero indicating a potential pole. Check the numerator, \(e^{2i} - 1\), which is non-zero. Hence, \(z = 2i\) is a simple pole (pole of order 1).

(b) \(\tan^{2}z, \ z=\frac{\pi}{2}\)

Notice that \(\tan(\frac{\pi}{2})\) does not exist as \(\tan z\) has a singularity at \(\frac{\pi}{2}\). The function \(\tan z\) has simple poles at \(\frac{\pi}{2} + n\pi\). Because \(\tan^2 z\) will have a pole of order 2 at \(z = \frac{\pi}{2}\).

(c) \(\frac{1-\cos z}{z^{4}}, \ z=0\)

Use Taylor series for cosine: \( 1 - \cos z = \frac{z^2}{2} - \frac{z^4}{24} + O(z^6)\). As \(z \to 0\), \( \frac{1-\cos z}{z^4} = \frac{\frac{z^2}{2} - \frac{z^4}{24} + O(z^6)}{z^4} = \frac{\frac{1}{2} - \frac{z^2}{24} + O(z^4)}{z^2}\) and dominant term at \(z = 0\) is \(\frac{1}{2z^2}\), hence \(z=0\) is a pole of order 2.

(d) \cos \left(\frac{\pi}{z-\pi}\right), \ z=\pi\)

Write \(\cos \left(\frac{\pi}{z-\pi}\right)\). As \(z \to \pi\), \ \frac{\pi}{z-\pi} \to \infty\. \cos(\infty)\ does not settle on a single value. Thus, \(z = \pi\) is an essential singularity.

Key concepts

Singularities

In complex analysis, a singularity is a point where a function is not defined or does not behave nicely. Singularities can be classified into three types:

• Pole
• Essential singularity
• Removable singularity

A removable singularity is a point where the function can be redefined to make it analytic. Poles and essential singularities are more complicated and cannot be 'fixed' by simple redefinition.

Poles

A pole is a type of singularity where the function goes to infinity as it approaches the point. Poles come in different orders, which describe how quickly the function diverges near the singularity. For example:

• Order 1: Simple Pole
• Order 2: Double Pole
• Order 3: Triple Pole

To determine the order of a pole, examine the function and see how many times you need to multiply by \(z - a\) (where \(a\) is the point in question) to get a non-zero limit as \(z \to a\). For instance, \(\frac{e^z - 1}{z^2 + 4}\) at \(z = 2i\) is a simple pole because multiplying once by \((z - 2i)\) yields a finite non-zero limit.

Essential Singularity

An essential singularity is a point where the function exhibits extremely erratic behavior as it approaches the point. This means that every neighborhood of the point includes values that are arbitrarily close to any number. Unlike poles, you can't 'tame' essential singularities by multiplying the function by \(z - a\).

A classic example is \(\text{cos}(\frac{\frac{\text{\text{\pi}}}{z - \pi}})\) at \(z = \pi\). As \(z \to \pi\), \(\frac{\text{\text{\pi}}}{z - \pi}\) becomes extremely large, causing \(\text{cos}(\frac{\text{\text{\pi}}}{z - \pi})\) to oscillate wildly between -1 and 1.

Regular Point

A regular point (or ordinary point) is a point where the function is analytic and behaves well. If a function has no singularity at a given point, it is considered regular at that point. Analyzing functions at regular points doesn’t introduce any complications, as the function behaves predictably and is differentiable in some neighborhood around the point.

Order of Poles

The order of a pole at a given point is the smallest positive integer \(n\) for which \((z - a)^n f(z)\) is analytic and non-zero at \(z = a\). In simpler terms, it describes how 'bad' the infinity behavior is near the pole:

• A pole of order 1 (simple pole) means the function approaches infinity linearly as it nears the point.
• A pole of order 2 (double pole) implies a quadratic behavior.
• Higher order poles mean even more rapid divergence.

For instance:\br>

• In \(\tan^2(z)\), \(z = \frac{\text{\text{\pi}}}{2}\)}\ has a pole of order 2.
• At \(z = 0\) in \(\frac{1 - \text{cos}(z)}{z^4}\), the pole is of order 2.