Question

Find series representations for all indicated regions. a. \(f(z)=\frac{z}{z-1},|z|<1,|z|>1\). b. \(f(z)=\frac{1}{(z-1)(2+2)},|z|<1,1<|z|<2,|z|>2\). [Hint Use partial fractions to write this as a sum of two functions first.]

Short answer

a) For \(f(z)=\frac{z}{z-1}\): For \(|z| < 1\), the series representation is \(1 + z + z^2 + z^3 + ...\). For \(|z| > 1\), it is \(-1/z - 1/z^2 - 1/z^3 - ...\). b) For \(f(z)=\frac{1}{(z-1)(2+z)}\): For \(|z| < 1\), it is \(-\frac{1}{3}(1 + z + z^2 + ... ) + \frac{1}{3}(1 - z/2 + (z/2)^2 - ...)\), For \(1 < |z| < 2\), it is \(-\frac{1}{3z}(1 + 1/z + 1/z^2 + ... ) + \frac{1}{3}(1 - z/2 + (z/2)^2 - ...)\), and for \(|z| > 2\), it is \(-\frac{1}{3z}(1 + 1/z + 1/z^2 + ... ) - \frac{1}{3z}(1 - 1/(z/2) + (1/(z/2))^2 - ...)\).

Step-by-step solution

Analyze function (a)

First, analyze the function \(f(z)=\frac{z}{z-1}\). This can be rewritten in a suitable form for a geometrical series expansion as \(f(z)=\frac{1}{1-z}\). This is the classic geometric series representation, where the common ratio is \(z\).

Determine series representation for function (a)

Using the formula of geometric series, \(\frac{1}{1-z} = 1 + z + z^2 + z^3 + ...\). Thus this becomes the series representation for \(|z| < 1\). For \(|z| > 1\), consider \(f(1/z) = \frac{1}{z(1-1/z)} = -1/z - 1/z^2 - 1/z^3 - ...\). This corresponds to the series representation over the region \(|z| > 1\).

Analyze function (b)

Next, analyze the function \(f(z)=\frac{1}{(z-1)(2+z)}\). Considering the hint, the aim is to decompose this fraction using partial fraction methodology. Expressing this as the sum of two simpler fractions gives us \(f(z) = \frac{A}{z-1} + \frac{B}{2+z}\). Calculating coefficients \(A\) and \(B\) provides \(A = -1/3\) and \(B = 1/3\).

Determine series representation for function (b) for the first region

Focus first on the region \(|z| < 1\). Thus, the series development results are \(f(z) = -\frac{1}{3}(1 + z + z^2 + ... ) + \frac{1}{3}(1 - z/2 + (z/2)^2 - ...)\).

Determine series representation for function (b) for the second region

Now, look at the region \(1 < |z| < 2\). Using the same approach, one can find the series representation as \(f(z) = -\frac{1}{3z}(1 + 1/z + 1/z^2 + ... ) + \frac{1}{3}(1 - z/2 + (z/2)^2 - ...)\).

Determine series representation for function (b) for the third region

Lastly, focus on the region \(|z| > 2\). The series representation here is \(f(z) = -\frac{1}{3z}(1 + 1/z + 1/z^2 + ... ) - \frac{1}{3z}(1 - 1/(z/2) + (1/(z/2))^2 - ...)\). In each region, the series representation is determined by ensuring that the modulus of the ratio in each term of the series is less than one.

Key concepts

Geometric Series Expansion

The geometric series expansion is a powerful tool in complex analysis for expressing functions as an infinite sum of terms. It is usually written in the form \( 1 + z + z^2 + z^3 + ... \), where \( z \) is a complex variable. For this to be valid, the absolute value of \( z \) must be less than one, denoted as \( |z| < 1 \).

When analyzing complex functions such as \( f(z)=\frac{z}{z-1} \) for the region \( |z|<1 \) we leverage the geometric series by rewriting the function as \( f(z)=\frac{1}{1-z} \), where \( 1-z \) then becomes the denominator with a form suitable for geometric expansion. The corresponding geometric series thus describes the function within the specified region. For regions where \( |z| > 1 \), the function needs to be manipulated, often by substituting \( z \) with \( 1/z \) to ensure the series expansion remains valid. This adjustment is crucial for the convergence of the series and the accurate representation of the function.

Complex Functions

Complex functions are mathematical expressions where the variables and constants are complex numbers. These functions can be analyzed and simplified using various techniques, including series expansion. The series representation breaks down these complex functions into simpler, infinite sums that are easier to manage and study.

In our example, the functions \( f(z)=\frac{z}{z-1} \) and \( f(z)=\frac{1}{(z-1)(2+z)} \) are initially complex due to their denominators. When looking at different regions based on the absolute value of \( z \), we see that the behavior of the function changes. By finding series representation in each region, we gain a more comprehensive understanding of the function and ensure that it is expressed in a form that allows for further analysis and application, such as integration or solving complex equations.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler ones that are easier to integrate or sum. It is particularly useful when dealing with complex functions that have multiple factors in the denominator.

Consider the function \( f(z)=\frac{1}{(z-1)(2+z)} \). The goal is to split this into a sum of simpler fractions. By using partial fractions, we represent \( f(z) \) as \( f(z) = \frac{A}{z-1} + \frac{B}{2+z} \), where \( A \) and \( B \) are constants to be determined. Once we calculate the appropriate values for \( A \) and \( B \) through algebraic manipulation, the resulting expression allows for a straightforward geometric series expansion about different ranges of \( z \). This decomposition is fundamental for developing series representations especially when the function's domain is divided into distinct regions, each requiring a different treatment for the series expansion.