91Ó°ÊÓ

In complex analysis, a **complex variable** is a variable whose values are complex numbers. A complex number has the form \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2=-1\). Complex variables are fundamental in many areas of mathematics and physics.

This exercise involves the function \(\frac{1}{1-z}\), which is analyzed for both \(|z| < 1\) and \(|z| > 1\). These regions refer to the magnitude of the complex variable \(z\) in the complex plane. When dealing with complex variables, it's important to consider:

• The real part and imaginary part of the variable
• The magnitude (or modulus), given by \(|z| = \sqrt{x^2 + y^2}\)
• The argument (or angle), denoted as \(\arg(z)\), which specifies the direction of the complex number

Understanding these aspects helps in the analysis and representation of functions involving complex variables.

A **series expansion** is the representation of a function as an infinite sum of terms. The Taylor series is a specific type of series expansion centered around a point (often zero). For instance, the Taylor series for \(\frac{1}{1-z}\) around \(z=0\) is:

\[\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n \ \text{for} \ |z| < 1\]

This series converges when the magnitude of \(z\) is less than 1. In contrast, the challenge in our exercise is to find a suitable series expansion for \(\frac{1}{1-z}\) when \(|z| > 1\).
The steps are:

• Rewriting the function with a useful substitution
• Using a known series expansion
• Substituting and simplifying expressions

This approach transforms the function and extends the series expansion to a new region of convergence.

**Analytic continuation** is a method to extend the domain of a given analytic (complex differentiable) function. It involves finding a new function that matches the original one within its initial domain.

For our series, the original Taylor series \(\frac{1}{1-z}\) expands around \(z=0\) for \(|z| < 1\). To continue this analytically for \(|z| > 1\), we:

• Reformulate the function in a manner suitable for the new domain
• Apply the series expansion to the transformed function
• Combine and simplify terms

The result is a new representation valid in the extended region. This process ensures the function retains its properties while becoming useful over a broader range of values.

The **radius of convergence** is the range within which a power series converges. For a series \(\sum_{n=0}^{\infty} a_n z^n\), the radius of convergence \(R\) is defined by:

\[\frac{1}{R} = \limsup_{n \rightarrow \infty} \sqrt[n]{|a_n|}\]

In simpler terms, it tells us the distance from the center (in this case, \(z=0\)) within which the series provides valid, convergent values.

In the case of \(\frac{1}{1-z}\) and its Taylor series around \(z=0\), the radius of convergence is 1, applicable for \(|z| < 1\).
Our task extends representation beyond this circle. For \(|z| > 1\), we transform the function using substitutions. The new series obtained, \(\sum_{n=0}^{\infty} z^{-(n+1)}\), converges for \(|z| > 1\), effectively shifting the radius of convergence to this region.

Understanding the radius of convergence is vital as it ensures the correctness and applicability of the series expansion.