For each of the following functions, say whether the...

For each of the following functions, say whether the indicated point is regular, an essential? singularity, or a pole, and if a pole of what order it is. (a) $\frac{\sin z-2}{z^{6}}, z=0$ (b) $\frac{z^{2}-1}{\left(z^{2}+1\right)^{2}}, z=i$ (c) $z e^{t / z}, \quad z=0$ (d) $\Gamma(z), \quad z=0)$ [See Chapter 11, equation (4.1).]

For each of the following functions, say whether the indicated point is regular, an essential? singularity, or a pole, and if a pole of what order it is.
(a) $\frac{\sin z-2}{z^{6}}, z=0$
(b) $\frac{z^{2}-1}{\left(z^{2}+1\right)^{2}}, z=i$
(c) $z e^{t / z}, \quad z=0$
(d) $\Gamma(z), \quad z=0)$
[See Chapter 11, equation (4.1).]

Key Concepts

Laurent Series

The Laurent series is an expansion of a complex function that includes both non-negative and negative powers. It is essential for analyzing the local behavior of functions near singular points and for classifying singularities as removable, poles, or essential singularities.

Regular (Analytic) Points

A regular point, or analytic point, is one where the function is well-behaved and has a convergent power series expansion. There is no singular behavior at these points, meaning the function is differentiable (holomorphic) in a neighborhood around the point.

Poles

A pole is an isolated singularity at which a function goes to infinity in a controlled manner. Near a pole, the function can be expressed as a Laurent series with a finite principal part consisting of a finite number of negative power terms.

Order of a Pole

The order of a pole is defined as the highest power of the negative exponent in the Laurent series expansion about the pole. If the Laurent expansion has terms up to 1/(z - a)^n and no higher negative powers, the point z = a is a pole of order n.

Essential Singularities

An essential singularity is an isolated point where the function exhibits wild behavior due to the presence of infinitely many negative power terms in its Laurent series. Unlike poles, the function does not tend toward infinity in a fixed manner near an essential singularity, and the behavior is described by the Casorati–Weierstrass theorem.

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