Introduction to Finding Residues in Complex Analysis

In the realm of complex analysis, the concept of residues plays a crucial role in understanding the behavior of functions near their singularities. This article focuses on finding the residues of the function f(z)1z2 i2z, where i is the imaginary unit. The function has two poles at z pm i, each of order 2.

Understanding the Function and Its Poles

The given function is f(z)1z2 i2z. Simplifying the denominator, we get f(z)1z2 z212z2.

The denominator 2z2 can be rewritten as 2z2, indicating a double pole at z pm i.

Using Partial Fraction Decomposition and Laurent Series Expansion

One method to find the residue is by using partial fraction decomposition and the Laurent series expansion. The function can be decomposed as:

f(z)i/4z-i -1/4z2 -i/4z i -1/4z2.

The coefficient of 1/z-a in the Laurent expansion is the residue at a. Thus, the residues at z i and z -i are Resfzi-i/4 and Resfz-ii/4, respectively.

Using the Contour Integration Formula

An alternative and my preferred method is to use the formula for residues of a pole of order n:

Resaf1n-1!limz→aOPTARGn-1->{_z_}(z-a)^n-1->{_z_}f(z).

Applying this formula to our function, we get:

Reszif12-1!limz→iOPTARG2-1->{_z_}(z-i)^1->{_z_}12z2.

Evaluating the limit and simplifying, we find the same result as before: Resfzi/4.

Visualizing the Behavior of the Function

When visualizing the behavior of the function f(z) near its singularities, we can see the Laurent series expansion helps in understanding the detailed behavior. For instance, near z i, the function behaves as follows:

f(z)12z2 -1/4 -i/4z-iz-i -1/4z2z2.

This Laurent series expansion clearly shows the role of each term, with the coefficient of 1/z-i giving the residue at z i.

Conclusion

Understanding the residues of a function is essential for many applications in complex analysis. By using partial fraction decomposition, the Laurent series expansion, and the contour integration formula, we can systematically find the residues and gain deeper insights into the behavior of complex functions near their singularities.

Keywords

residue, Laurent series, complex analysis, pole, function singularity