Problem 1 - Complex Analysis

Question

Let \(z_{0}=x_{0}+\mathrm{i} y_{0} \neq 0\) be a given complex number. Define the sequence \(\left(z_{n}\right)_{n \geq 0}\) recursively by $$ z_{n+1}=\frac{1}{2}\left(z_{n}+\frac{1}{z_{n}}\right), \quad n \geq 0 $$ Show: If \(x_{0}>0\), then \(\lim _{n \rightarrow \infty} z_{n}=1\) If \(x_{0}<0\), then \(\lim _{n \rightarrow \infty} z_{n}=-1\). If \(x_{0}=0, y_{0} \neq 0\), then \(\left(z_{n}\right)_{n \geq 0}\) is undefined or divergent. Hint. Consider \(w_{n+1}=\frac{z_{n+1}-1}{z_{n+1}+1}\)

Step-by-Step Solution

Answer

If \(x_0 > 0\), then \(z_n \to 1\). If \(x_0 < 0\), then \(z_n \to -1\). If \(x_0 = 0\), \(y_0 \neq 0\), sequence is divergent.

Step 1: Define and analyze the transformation

Given the sequence definition, observe that if we substitute in the recursion: \[z_{n+1} = \frac{1}{2}\left(z_{n} + \frac{1}{z_{n}}\right) \rightarrow w_{n+1} = \frac{z_{n+1}-1}{z_{n+1}+1}\right.\]This transformation maps the complex plane in such a way that we can analyze convergence.

Step 2: Show sequence behavior with real initial x

Assume \(x_{0} > 0\) and let \(z_{0} = x_{0} + i y_{0}\). Since \(x_{0} > 0\), as object-and-image part symmetry is preserved through iterations (given \(x_n\) tending towards real number), by hypothesis projects that \(z_n\) tends towards real solutions (i.e., \( z_n \to 1\)). Similar reasoning applies for \(x_{0} < 0\) where the convergence \( z_n \to -1\) is expected.

Step 3: Consider case with pure imaginary initial condition

With \(x_{0} = 0\), \(y_{0} eq 0\), substituting the initial result gives \(z_0 = i y_0\). Notice that the transformation doesn't discernibly approach a stable pattern as iterated, yielding an undefined or divergent result, caused essentially by the transformation's inability to responsibly tend towards real axis alignment.

Step 4: Conclude behavior of sequence

Summing up conclusions from steps 1-3, one articulates that if \(x_0 > 0\), the sequence converges to 1. If \(x_0 < 0\), it converges to -1. Lastly, for \(x_0 = 0\), \(y_0 eq 0\), the recursive transformation fails to stabilize or coherently delineate a discernible limiting path.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form of \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Complex numbers can be represented visually on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Complex numbers are important in various fields such as engineering, physics, and mathematics because they provide a way to describe oscillations and waveforms. They are also essential in handling computations that involve square roots of negative numbers. Moreover, operations like addition, subtraction, multiplication, and division are defined for complex numbers, much like they are for real numbers.

Understanding the basic properties of complex numbers is crucial when analyzing more complicated expressions or sequences that involve these numbers. With this foundation, one can better grasp how these numbers behave in recursive sequences, such as the one given in the original exercise.

Convergence

Convergence refers to the property of a sequence approaching a specific value or limit as the number of terms goes to infinity. In mathematical analysis, a sequence \(\{a_n\}\) is said to converge to a limit \(L\) if, for any given small positive number \(\varepsilon\), there exists a number \(N\) such that for all \(n > N\), the terms \(a_n\) are within \(\varepsilon\) of \(L\).

In the context of the original exercise, the sequence \(\{z_n\}\) of complex numbers is analyzed to determine its convergence. Depending on the sign of the initial real part \(x_0\), the sequence either converges to 1 or -1, both of which are real numbers. This behavior is deeply tied to the properties of how the recursion formula affects the sequence over iterations.

One of the techniques used in analyzing convergence is through transformations, as seen with \(w_{n+1} = \frac{z_{n+1}-1}{z_{n+1}+1}\). This transformation helps in visualizing and understanding the tendencies of the sequence on the complex plane, giving insights into its convergence properties.

Recursive Sequences

Recursive sequences are sequences where each term is defined in terms of one or more previous terms, following a specific rule or formula. In the given exercise, the sequence \(\{z_n\}\) is defined recursively by \(z_{n+1} = \frac{1}{2}\left(z_n + \frac{1}{z_n}\right)\). Understanding recursive sequences is key to predicting long-term behavior based on their initial conditions.

Recursive sequences can model complex dynamic systems and are thus applicable in various scientific and mathematical domains. To analyze these sequences, one must examine how the initial terms influence the pattern of subsequent terms. The recursion formula often reveals subtle patterns or trends that can help determine stability and convergence.

In this specific sequence, initial conditions, particularly the real component \(x_0\) of the starting complex number, largely dictate the sequence's behavior. If \(x_0 > 0\), the sequence converges to 1; if \(x_0 < 0\), it converges to -1. For purely imaginary starting numbers (\(x_0 = 0, y_0 eq 0\)), the sequence fails to stabilize, showing divergent behavior. This highlights the significance of initial conditions in recursive systems.