Question

\(^{\star}\) Bestimme die Art der Singularität der Funktion 426 8 Isolierte Singularitäten, Laurent-Entwicklung $$ f(z)=\left(z^{2}+2\right) \sin \frac{1}{z-1} $$ und entwickle \(f\) in eine Laurentreihe um ihre Singularität.

Short answer

The function \(f(z) = (z^{2}+2) \sin \frac{1}{z-1}\) has an essential singularity at \(z = 1\), and the Laurent series development around this singularity is \(f(z) = (z^{2}+2) \left( \frac{1}{z-1} - \frac{1}{6(z-1)^3} + \frac{1}{120(z-1)^5} - \ldots\right)\).

Step-by-step solution

Find the singularities

We start by identifying singularities from the function, which are values that would make \(\frac{1}{z-1}\) undefined. Here, the singular point is \(z=1\).

Determine type of singularity

Next, we need to find out what kind of singularity we have at \(z=1\). An essential singularity is a complex version of the discontinuity. It has a condition that a limit does not tend to any finite value. In this case, the function \(\sin \frac{1}{z-1}\) has an essential singularity at \(z=1\). The factor \(z^{2}+2\) is analytic (i.e., has no singularities), thus, doesn't change the singularity type of the function. Therefore, \(f(z)\) has an essential singularity at \(z=1\).

Laurent series development

The Laurent series of \(f(z)\) around \(z=1\) is developed by first considering the series expansion for \(\sin\) function, \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots\). Now for the replacement \(x \rightarrow \frac{1}{z-1}\), the Laurent series expansion of \(f(z)\) is \(f(z) = (z^{2}+2) \left( \frac{1}{z-1} - \frac{1}{6(z-1)^3} + \frac{1}{120(z-1)^5} - \ldots\right)\)

Key concepts

Isolated Singularities

In complex analysis, understanding isolated singularities is essential for analyzing certain complex functions. These are points in the complex plane where a function behaves abnormally. An isolated singularity at a point, say \( z = a \), means the function is not defined, or not analytic, at \( z = a \), but defined everywhere else in some neighborhood around \( a \). It is crucial to recognize these points to further classify the type of singularity and its impact on the function.

• If you can identify a point where the function fails to be analytic, such as \( z = 1 \) in our function \( f(z) = \left(z^{2}+2\right) \sin \frac{1}{z-1} \), you've found an isolated singularity.
• Analyzing what happens at these singularities gives insight into the function's behavior around the complex plane.
• Isolated singularities are the simplest form of complex singularities and are fundamental in understanding the more complex types.

Isolated singularities pave the way to exploring the next aspects like Laurent series or essential singularities.

Laurent Series

The Laurent series is a representation of a complex function similar to the Taylor series. However, it includes terms with negative powers, which are critical for capturing the behavior of functions with singularities. The Laurent series expansion is particularly useful for functions with isolated singularities, as seen in \( f(z) = \left(z^{2}+2\right) \sin \frac{1}{z-1} \).

• A standard Taylor series can only represent functions that are analytic at the expansion point. In contrast, a Laurent series can handle singularities by accommodating negative powers.
• The general form of a Laurent series is:\[\sum_{{n=-fty}}^{fty} a_n (z - a)^n\]where \( a_n \) are coefficients determined by the function, and \( a \) is the point around which the expansion is done.
• In our example, the series is generated at \( z = 1 \) using: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\]substituting \( x = \frac{1}{z-1} \), yields terms with negative exponents revealing more about the function structure.

The Laurent series is vital in classifying singularities and studying complex functions.

Essential Singularities

Essential singularities examine the behavior of complex functions at their "wildest" points. These are singularities where not only does a function not behave regularly, but its behavior defies all finite limits. For \( f(z) = \left(z^{2}+2\right) \sin \frac{1}{z-1} \), the point \( z = 1 \) is an essential singularity due to the \( \sin \frac{1}{z-1} \) component.

• Unlike poles or removable singularities, essential singularities result in a function that takes nearly every possible value in any nearby neighborhood, albeit within a single limit point.
• Entirely unusual, essential singularities are characterized by the fact that neither the limit nor the value approaches anything consistent, as the case with our function.
• In many instances, essential singularities demonstrate dramatic transformations in function behavior, underscored by methods like the Laurent series.

Describing essential singularities provides a comprehensive understanding of some of the most erratic behaviors in complex analysis.