Problem 17 - Advanced Engineering Mathematics

Question

In Problems 13-24, determine the order of the poles for the given function. $$ f(z)=\tan z $$

Step-by-Step Solution

Answer

The function \( f(z) = \tan z \) has simple poles (order 1) at \( z = \frac{\pi}{2} + k\pi \).

Step 1: Definition of a Pole

A pole of a function is a type of singularity where the function goes to infinity. To find poles of the function \( f(z)=\tan z \), we first need to identify where \( \tan z \) becomes undefined.

Step 2: Identifying Poles of \(\tan z\)

The tangent function, \( \tan z = \frac{\sin z}{\cos z} \), is undefined wherever \( \cos z = 0 \). The cosine function is zero at \( z = \frac{\pi}{2} + k\pi \), where \( k \) is any integer. Hence, the poles of \( \tan z \) are at these points.

Step 3: Determine the Order of Each Pole

The order of a pole is the smallest positive integer \( n \) such that \( (z-z_0)^n f(z) \) is analytic (holomorphic) at \( z=z_0 \). Since \( \cos z \) has a simple zero at \( z = \frac{\pi}{2} + k\pi \), the pole of \( \tan z \) at this point is of order 1 (simple pole).

Step 4: Conclusion

Thus, the function \( f(z) = \tan z \) has simple poles at \( z = \frac{\pi}{2} + k\pi \).

Key Concepts

Singularity in Complex Analysis

In complex analysis, singularities are points at which a complex function does not behave normally. They are critical in understanding complex functions' behaviors. Every singularity offers important insights into the nature of a function. There are different types of singularities, including removable singularities, poles, and essential singularities. Each type offers unique characteristics. For instance, a removable singularity can be "fixed" by extending the function to include the singularity point. In contrast, poles are points where a function heads towards infinity. Discussing poles, specifically, will lead us into understanding the concept of their order and classification.

Order of Poles

The order of a pole tells us about the behavior of a function around a singularity. To define it more clearly, imagine a function becomes infinite at a certain point. The order of a pole is the smallest positive integer that, when multiplied by a corresponding power of the difference from that point, results in an analytic function. For example, if a function behaves like \(\frac{1}{(z-z_0)}\), it has a simple pole of order 1 at point \(z_0\). If it behaves like \(\frac{1}{(z-z_0)^2}\), then it has a pole of order 2, and so on. Determining the order is crucial as it affects the residue calculation and hence other aspects of complex integration.

Complex Analysis

Complex Analysis is a fascinating field of mathematics focused on functions of complex variables. Unlike real analysis, complex analysis offers deeper insights because it leverages the rich structure of complex numbers. This branch of mathematics explores mappings, linearity, and transformations that are not visible in real-numbered functions. Concepts like contour integration, residues, and conformal mappings make complex analysis distinct. In understanding poles and singularities, complex analysis provides powerful tools to analyze and predict function behaviors over complex planes. This makes it very applicable in diverse fields such as physics and engineering.

Tangent Function

The tangent function, denoted as \(\tan z\), is a fundamental trigonometric function that extends into the realm of complex numbers. It is defined as the ratio of the sine function to the cosine function: \(\tan z = \frac{\sin z}{\cos z}\). While simple in its real form, its behavior in the complex domain requires careful exploration. The tangent function encounters undefined behavior, or poles, wherever the cosine function hits zero. These poles are recurring at points like \((z = \frac{\pi}{2} + k\pi)\). Understanding where the function tends to infinity allows for precise mapping of its behavior in complex planes, aiding broader complex analysis.