Understanding the Residue of the Function f(z) z^2 / (z^2 - 1)2 at z-2

When dealing with complex analysis, the behavior of functions around certain points is a crucial concept to grasp. A particular point in the complex plane where a function exhibits interesting behavior, such as a pole or an essential singularity, plays a significant role in understanding the function's overall structure. In this article, we will delve into the residue of the function f(z) z^2 / (z^2 - 1)2 at z -2. We will explore the nature of this point and how to compute the residue using limits.

Introduction to Complex Functions

In the realm of complex analysis, functions of a complex variable (complex functions) take on a unique significance. These functions can have singularities, which are points where the function is not analytic. A simple pole is a type of singularity where the function behaves like 1/(z - a) near the point a. If the function has a higher-order pole, it behaves like (z - a)^-k where k > 1.

The Function and Its Poles

The function in question is f(z) z^2 / (z^2 - 1)2. To analyze the behavior of this function near z -2, we first need to identify the poles of the function. A pole is a point where the function is not defined due to a zero in the denominator. In the given function, the denominator is (z^2 - 1)2, which simplifies to (z - 1)2(z 1)2.

At z 1 and z -1, the function has poles of order 2. However, we are specifically interested in the point z -2. To determine the behavior of the function at z -2, we need to recognize that the denominator does not have a zero at z -2. Instead, we need to rewrite the function in a form that highlights the behavior near z -2.

Poles and Residues

A residue is a coefficient in the Laurent series expansion of a function around a pole. For a simple pole (pole of order 1), the residue can be calculated using the limit formula: res(f, a) limz→a(z - a) f(z). For a higher-order pole, the process is more involved, but the basic principle remains the same.

Determining the Nature of the Pole

In the function f(z) z^2 / (z^2 - 1)2, the point z -2 is not a pole of the function. However, to compute the residue at a point where the function has a pole, we can still use the limit formula. The function can be simplified for the calculation as follows:

limz→-2 (z 2) * (z^2 / (z^2 - 1)2)

First, we need to expand (z^2 - 1)2:

(z^2 - 1)2 (z^2 - 1)(z^2 - 1) z^4 - 2z^2 1

Now, let's rewrite the function and apply the limit:

Calculating the Residue

We need to find:

res(f, -2) limz→-2 (z 2) * (z^2 / (z^4 - 2z^2 1))

Simplifying further:

res(f, -2) limz→-2 (z^2 / (z^2 - 1)^2)

Now, substitute z -2 into the simplified expression:

res(f, -2) (-2)2 / ((-2)2 - 1)^2 4 / (4 - 1)^2 4 / 9

Hence, the residue of the function f(z) z^2 / (z^2 - 1)2 at z -2 is 4/9.

Conclusion

The residue of a function at a point where it has a pole of order k is an essential concept in complex analysis. In this article, we explored the function f(z) z^2 / (z^2 - 1)2, identified the point z -2 as a non-pole, and calculated the residue at this point using limits. The residue is a fundamental tool in complex integration and has wide-ranging applications in both theoretical and applied mathematics.