Problem 19 - A First Course in Complex Analysis with Applications

Question

Determine the order of the poles for the given function. $$ f(z)=\tan z $$

Step-by-Step Solution

Answer

Each pole of \( \tan z \) is of order 1.

Step 1: Identify the Poles of the Function

The function \( f(z) = \tan z \) is the quotient of sine and cosine: \( \tan z = \frac{\sin z}{\cos z} \). To find the poles, we need to find the values of \( z \) where the denominator, \( \cos z \), equals zero.

Step 2: Solve for the Zeros of the Denominator

The cosine function, \( \cos z \), equals zero at \( z = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. These are the values of \( z \) where \( \cos z = 0 \), giving the locations of the poles of \( f(z) \).

Step 3: Determine the Order of Each Pole

For each pole, such as at \( z = \frac{\pi}{2} \), we examine the behavior of \( \cos z \) around this point. Since \( \cos z \) has a simple zero at these points, the function \( \frac{1}{\cos z} \) has a simple pole there. Therefore, each pole of \( \tan z \) is of order 1.

Key Concepts

Poles of Complex Functions

In complex analysis, poles are crucial points where a complex function ceases to be analytic, meaning it cannot be expressed as a Taylor series. To find poles in a complex function, observe where the denominator of a complex fraction becomes zero while the numerator remains analytic. These points hint at poles because the function becomes undefined.
For instance, consider the function \( f(z) = \tan z \), expressed as:

• \( \tan z = \frac{\sin z}{\cos z} \)

Here, poles occur where \( \cos z = 0 \), as the function cannot maintain its value when divided by zero. These critical points of the denominator determine the locations of the poles for the function. Identifying poles helps in analyzing function behavior and potential singularities, preparing for further study such as residue calculation.

Trigonometric Functions in Complex Analysis

Trigonometric functions are fundamental in both real and complex analysis due to their periodic nature and elegant properties. In complex analysis, functions like sine, cosine, and tangent offer unique perspectives and wider scopes. They often reveal the richness of analysis when complex variables are used.
Take \( \tan z\), the ratio of \( \sin z \) to \( \cos z \). When handled over complex numbers, trigonometric functions undergo transformations, laying out complex zeroes and poles. For well-known trigonometric components:

• \( \sin z \) and \( \cos z \) are periodic with periods \( 2\pi \).
• The zeroes of \( \cos z \) determine where \( \tan z \)'s poles occur.

Understanding these trigonometric functions and their behaviors in the complex plane is foundational for exploring deeper mathematical principles and applications.

Order of Poles

The order of a pole tells us about the behavior of a function near its pole. A pole of order 1, often called a simple pole, indicates that as the complex variable nears the pole, the function behaves like \( \frac{1}{z-z_0} \), where \( z_0 \) is the pole location.
For \( f(z) = \tan z \), because \( \cos z \) has simple zeros (each zero is of multiplicity one), each pole at \( z = \frac{\pi}{2} + n\pi \) for any integer \( n \), is of order 1. Determining the order signifies how steeply the function approaches infinity near the pole, impacting residue and integral computations within complex contour integrals.
Effectively, knowing the order aids in predicting curve behavior and helps articulate the function's analytic characteristics across its domain.