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

The sequence converges to 1 if \(x_0 > 0\), to -1 if \(x_0 < 0\), and is divergent or undefined if \(x_0 = 0, y_0 \neq 0\).

Step 1: Introducing the Transformation

To analyze the sequence, consider the transformation given by the hint: \(w_{n+1} = \frac{z_{n+1} - 1}{z_{n+1} + 1}\). This transformation can help in understanding the behavior of \(z_n\) as \(n\) approaches infinity.

Step 2: Derive the Recurrence for wn

Express \(z_{n+1}\) in terms of \(w_n\). Start by substituting \(z_n = 1 + \frac{2w_n}{1-w_n}\) back into \(z_{n+1} = \frac{1}{2}(z_n + \frac{1}{z_n})\). After simplification, derive the relation \(w_{n+1} = \frac{w_n}{1+w_n^2}\).

Step 3: Analyze the Behavior of wn

Check the limit of \(w_n\) using the derived recurrence. As \(n\) tends towards infinity, if \(w_n\) converges to zero, \(z_n\) will converge to \(1\). If \(w_n\) diverges or oscillates, \(z_n\) will not have a limit.

Step 4: Consider cases based on x0

- If \(x_0 > 0\): Initially \(w_0\) is a small real number, thus the recurrent relation will lead \(w_n\) to zero, implying \(z_n\) approaches\(1\).- If \(x_0 < 0\): \(w_0\) starts close to \(-1\), but similarly it converges to zero because \(w_{n+1} = \frac{w_n}{1+w_n^2}\) reduces each time.- If \(x_0 = 0, y_0 eq 0\): \(z_n\) becomes imaginary at some stage, causing the sequence to become undefined or diverge due to oscillations in \(w_n\).

Step 5: Conclude

Synthesize these results: - For \(x_0 > 0\), \(\lim_{n \to \infty} z_n = 1\). - For \(x_0 < 0\), \(\lim_{n \to \infty} z_n = -1\). - For \(x_0 = 0, y_0 eq 0\), the sequence is undefined or diverges.

Key Concepts

Complex Analysis

In the realm of mathematics, *Complex Analysis* is the study of complex numbers and functions of a complex variable. Complex numbers are numbers that include both a real part and an imaginary part, expressed in the form \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit, \( i \), is defined by \( i^2 = -1 \).

Complex analysis plays a pivotal role in various fields of engineering and physical sciences, offering fascinating insights, especially when dealing with sequences and functions in the complex plane. The concept of a *sequence* in complex analysis is not different from regular sequences, but it considers each term as a complex number.

• **Complex Plane:** The complex number \( z \) is represented on a two-dimensional plane, with the horizontal axis as the real part (x-axis) and the vertical axis as the imaginary part (y-axis).
• **Complex Functions:** Functions that take complex numbers as inputs and produce complex numbers as outputs, which are essential in solving problems related to the convergence and behavior of sequences and series.

This foundational knowledge is critical when exploring problems involving recursive sequences built from complex numbers, as demonstrated in the original exercise.

Recursive Sequences

A *Recursive Sequence* is a sequence of numbers where each subsequent term is determined by applying a formula to one or more of the preceding terms. The recursive nature causes the sequence to evolve step-by-step. In mathematics, especially complex analysis, understanding recursive sequences can help uncover larger patterns within seemingly random collections of terms.

In the provided exercise, the sequence \( (z_n)_{n \geq 0} \) is defined recursively by \( z_{n+1} = \frac{1}{2}(z_n + \frac{1}{z_n}) \). This means each term \( z_{n+1} \) is calculated by manipulating the preceding term \( z_n \), specifically using both the term itself and its reciprocal.

• **Base Case:** The given \( z_0 = x_0 + iy_0 \) provides the starting point for the sequence. Its value influences the behavior of the entire sequence.
• **Recursive Formula:** A mathematical formula that relates each term in the sequence to its immediate predecessor. In recursion, this is often a critical tool in solving limits and proving divergence or convergence.

The recursion allows us to explore the sequence's behavior under varying conditions of initial terms, leading to different convergence behaviors.

Convergence of Sequences

The *Convergence of Sequences* discusses whether a sequence approaches a specific value as the number of terms goes to infinity. This concept is central in mathematics for determining the ultimate stability or instability of a sequence.

For complex sequences, convergence involves both real and imaginary components, requiring the sequence's terms to cluster around a particular complex number in the plane. It is essential to consider the real part and the imaginary part while analyzing the convergence of complex sequences.

• **Convergent Sequence:** A sequence whose terms become arbitrarily close to a specific value, called the limit, as the number of terms grows.
• **Divergent Sequence:** A sequence that does not settle toward any particular value.

In the exercise:
- If \( x_0 > 0 \), the sequence converges to \( 1 \). - If \( x_0 < 0 \), it converges to \(-1\).- If \( x_0 = 0 \) with \( y_0 eq 0 \), the sequence does not converge, showcasing a typical divergence when initial terms constrain the sequence's ability to settle.

In practical terms, knowing how and why a sequence converges, particularly within complex analysis, provides foundational insights useful in engineering, physics, and even in financial modeling.