Problem 22 - A First Course in Complex Analysis with Applications

Question

Determine the order of the poles for the given function. $$ f(z)=\frac{e^{z}}{z^{2}} $$

Step-by-Step Solution

Answer

The pole at \( z=0 \) is of order 2.

Step 1: Identify the Denominator

To determine the poles of the function \( f(z) = \frac{e^z}{z^2} \), we first consider where the denominator is equal to zero. Since the denominator is \( z^2 \), it is equal to zero when \( z = 0 \). This indicates there is a pole at \( z = 0 \).

Step 2: Determine the Order of the Pole

The order of a pole is given by the highest power of \( z \) that divides the denominator and for which the numerator does not nullify the pole. In \( f(z) = \frac{e^z}{z^2} \), the term \( z^2 \) in the denominator indicates a potential order of 2. Since \( e^z = 1 + z + \frac{z^2}{2!} + \ldots \), the series starts from a non-zero constant term, hence it does not nullify the order of the pole.

Key Concepts

Poles of complex functions

In the realm of complex analysis, a pole of a complex function is a specific type of singularity. Poles are points in the complex plane where a function becomes unbounded, essentially indicating where the function blows up to infinity.
For example, consider the function \( f(z) = \frac{e^z}{z^2} \). In this scenario, examining the poles involves looking at the denominator, \( z^2 \), as it dictates where the function becomes undefined.
Here's how you can identify poles in complex functions:

• Check where the denominator equals zero — these are the potential poles.
• Ensure that the numerator doesn't also vanish at these points, as this might indicate a removable singularity instead.

For \( f(z) \), setting the denominator \( z^2 = 0 \) reveals a pole at \( z = 0 \). It's important to verify that this results in a true pole by confirming the numerator, \( e^z \), doesn't also vanish at \( z = 0 \). In this case, \( e^z \) remains non-zero, confirming a genuine pole existence.

Order of a pole

The order of a pole provides deeper insights into the behavior of a function near its poles. When a pole is identified, determining its order is the subsequent step. The order of a pole is defined by the power of \( z \) in the denominator that cannot be canceled by the numerator.
Consider our function \( f(z) = \frac{e^z}{z^2} \). Here, the potential pole at \( z = 0 \) is primarily due to the \( z^2 \) factor.
As the numerator \( e^z \) can be expanded as a Taylor series:
\[ e^z = 1 + z + \frac{z^2}{2!} + \ldots \]
we notice that the series starts with a non-zero constant term (i.e., 1). Therefore, it doesn't nullify the \( z^2 \) in the denominator, establishing the order as 2.
**Key Steps for Finding Order of a Pole:**

• Identify the power of \( z \) in the denominator.
• Ensure the numerator does not cancel this power out.

By following this logic, you solidify how poles alter the nature of complex functions.

Denominator analysis

Analyzing the denominator of a complex function is crucial in identifying and understanding singularities such as poles. Let's see how this applies to our function \( f(z) = \frac{e^z}{z^2} \).
The denominator of \( f(z) \) is \( z^2 \), which gives us a clear indication of where potential issues might arise. By setting the denominator equal to zero, \( z^2 = 0 \), we see that \( z = 0 \) is where the function is undefined, thus indicating a pole.
This pole is essential to the behavior of the entire function in its neighborhood.

• Check simple denominators by setting them to zero to find roots (potential poles).
• For complicated denominators, factorize or find roots systematically.

Understanding the nature of the denominator is instrumental, as it determines key characteristics of the function's graph and the behavior around infinity.
Comprehensive denominator analysis ensures accurate identification of all poles and their orders, thus aiding in the full understanding of complex functions.