Exploring Laurent Series: A Tale of Two Domains

Let’s dive into the world of complex analysis, where seemingly simple functions like 

f(z)=1z(1+z2)f(z) = \frac{1}{z(1+z^2)} reveal fascinating secrets when expressed as Laurent series. In this post, we’ll unravel two such series, each tailored for a specific domain in the complex plane. If you’re ready, let’s explore how to break down f(z)f(z) and discover its hidden structure step by step.

Step 1: What’s a Laurent Series?

A Laurent series is like a cousin to the Taylor series but with a twist. While a Taylor series represents functions as an infinite sum of powers of zz, a Laurent series expands functions using powers of zz and negative powers of zz. This makes Laurent series perfect for studying functions with singularities (places where the function "blows up" or becomes undefined).

For the function f(z)=1z(1+z2)f(z) = \frac{1}{z(1+z^2)}, there’s a singularity at z=0z = 0 because of the 1z\frac{1}{z} term and singularities at z=±iz = \pm i because 1+z2=01+z^2 = 0 at these points.

Our goal is to find the Laurent series expansions of f(z)f(z) in two distinct domains:

  1. Domain 1: ∣z∣<1|z| < 1
  2. Domain 2: ∣z∣>1|z| > 1

Step 2: Rewriting f(z)f(z)

The function is given by:

f(z)=1z(1+z2)f(z) = \frac{1}{z(1+z^2)}

We’ll rewrite it as:

f(z)=1z⋅11+z2f(z) = \frac{1}{z} \cdot \frac{1}{1+z^2}

The term 11+z2\frac{1}{1+z^2} can be expanded differently depending on whether ∣z∣<1|z| < 1 or ∣z∣>1|z| > 1, using the geometric series expansion 11−w=∑n=0∞wn\frac{1}{1-w} = \sum_{n=0}^\infty w^n (valid when ∣w∣<1|w| < 1).

Step 3: Laurent Series for ∣z∣<1|z| < 1

In this domain, ∣z∣<1|z| < 1, so ∣z2∣<1|z^2| < 1. This allows us to expand 11+z2\frac{1}{1+z^2} as a geometric series:

11+z2=∑n=0∞(−1)nz2n\frac{1}{1+z^2} = \sum_{n=0}^\infty (-1)^n z^{2n}

Substituting this into f(z)f(z), we get:

f(z)=1z⋅∑n=0∞(−1)nz2nf(z) = \frac{1}{z} \cdot \sum_{n=0}^\infty (-1)^n z^{2n}

Simplify the terms:

f(z)=∑n=0∞(−1)nz2n−1f(z) = \sum_{n=0}^\infty (-1)^n z^{2n-1}

And there you have it! The Laurent series for f(z)f(z) in ∣z∣<1|z| < 1 is:

f(z)=∑n=0∞(−1)nz2n−1f(z) = \sum_{n=0}^\infty (-1)^n z^{2n-1}

Step 4: Laurent Series for ∣z∣>1|z| > 1

Now, let’s explore ∣z∣>1|z| > 1. Here, we’ll rewrite 11+z2\frac{1}{1+z^2} to suit this domain. Notice that z2z^2 is large when ∣z∣>1|z| > 1, so we factor z2z^2 out of 1+z21+z^2:

1+z2=z2(1+1z2)1+z^2 = z^2 \left( 1 + \frac{1}{z^2} \right)

Thus:

11+z2=1z2⋅11+1z2\frac{1}{1+z^2} = \frac{1}{z^2} \cdot \frac{1}{1 + \frac{1}{z^2}}

Now, expand 11+1z2\frac{1}{1 + \frac{1}{z^2}} as a geometric series (valid because ∣1z2∣<1\left| \frac{1}{z^2} \right| < 1 when ∣z∣>1|z| > 1):

11+1z2=∑n=0∞(−1z2)n\frac{1}{1 + \frac{1}{z^2}} = \sum_{n=0}^\infty \left( -\frac{1}{z^2} \right)^n

Substituting this back, we have:

11+z2=1z2⋅∑n=0∞(−1)n1z2n\frac{1}{1+z^2} = \frac{1}{z^2} \cdot \sum_{n=0}^\infty (-1)^n \frac{1}{z^{2n}} 11+z2=∑n=0∞(−1)n1z2n+2\frac{1}{1+z^2} = \sum_{n=0}^\infty (-1)^n \frac{1}{z^{2n+2}}

Now substitute into f(z)=1z⋅11+z2f(z) = \frac{1}{z} \cdot \frac{1}{1+z^2}:

f(z)=1z⋅∑n=0∞(−1)n1z2n+2f(z) = \frac{1}{z} \cdot \sum_{n=0}^\infty (-1)^n \frac{1}{z^{2n+2}}

Simplify the terms:

f(z)=∑n=0∞(−1)n1z2n+3f(z) = \sum_{n=0}^\infty (-1)^n \frac{1}{z^{2n+3}}

Or equivalently:

f(z)=∑n=0∞(−1)nz−(2n+3)f(z) = \sum_{n=0}^\infty (-1)^n z^{-(2n+3)}

This is the Laurent series for f(z)f(z) in ∣z∣>1|z| > 1.

Step 5: Summary

To recap, we found two Laurent series for f(z)=1z(1+z2)f(z) = \frac{1}{z(1+z^2)}:

  1. For ∣z∣<1|z| < 1:
f(z)=∑n=0∞(−1)nz2n−1f(z) = \sum_{n=0}^\infty (-1)^n z^{2n-1}
  1. For ∣z∣>1|z| > 1:
f(z)=∑n=0∞(−1)nz−(2n+3)f(z) = \sum_{n=0}^\infty (-1)^n z^{-(2n+3)}

Each series captures the behavior of f(z)f(z) in its respective domain, showcasing the beauty and versatility of Laurent expansions.

Why It Matters

Laurent series are powerful tools for studying complex functions, especially near singularities. By expanding f(z)f(z) in these two domains, we gain insight into how the function behaves differently depending on where we are in the complex plane.

Thank you for joining this journey into the heart of complex analysis! If you have questions or want to see more examples, feel free to leave a comment. Sweet dreams of Laurent series!

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