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Magazine Chapter 17: Problem 12
Find the fixed points. $$w=\frac{z-1}{z+1}$$
Short Answer
Expert verified
The fixed points are \( z = i \) and \( z = -i \).
Step by step solution
01
Understand the Problem
A fixed point of a function is a point that is mapped to itself by the function. In this case, we want to find the values of \( z \) such that \( w = z \), where \( w = \frac{z-1}{z+1} \).
02
Set the Function Equal to Its Input
Set \( w = z \) in the given function: \( z = \frac{z-1}{z+1} \). Our goal is to solve this equation for the values of \( z \).
03
Clear the Fraction by Cross-Multiplying
To eliminate the fraction, cross-multiply: \( z(z + 1) = z - 1 \). This gives: \( z^2 + z = z - 1 \).
04
Simplify the Equation
Subtract \( z \) from both sides to simplify the equation: \( z^2 + z - z = z - z - 1 \). This simplifies to \( z^2 = -1 \).
05
Solve for Fixed Points
To solve \( z^2 = -1 \), find the values of \( z \). Taking the square root of both sides, we get \( z = i \) or \( z = -i \), where \( i \) is the imaginary unit.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Complex Functions
Complex functions are mathematical expressions that involve complex numbers. A complex number is composed of a real part and an imaginary part, commonly written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part with \(i = \sqrt{-1}\).
Complex functions extend real functions into the complex domain, allowing for a broader set of solutions and behaviors. They play a crucial role in various fields, including engineering, physics, and applied mathematics. In the given exercise, we deal with the function \(w = \frac{z-1}{z+1}\), where both \(w\) and \(z\) can be complex numbers.
Complex functions extend real functions into the complex domain, allowing for a broader set of solutions and behaviors. They play a crucial role in various fields, including engineering, physics, and applied mathematics. In the given exercise, we deal with the function \(w = \frac{z-1}{z+1}\), where both \(w\) and \(z\) can be complex numbers.
- Understand that \(z\) can take any complex value.
- Fixed points of complex functions are where the function's output equals its input.
Cross-Multiplication Simplified
Cross-multiplication is often used to eliminate fractions in equations, simplifying the solving process. To cross-multiply, multiply the numerator of each fraction by the denominator of the other fraction. This is particularly helpful when equations involve rational expressions.
In our exercise, we set \(z = \frac{z-1}{z+1}\) and cross-multiply to eliminate the fraction:
In our exercise, we set \(z = \frac{z-1}{z+1}\) and cross-multiply to eliminate the fraction:
- Multiply \(z\) by \((z+1)\).
- Multiply \(z-1\) by 1 (since there is no denominator on the left, it can be treated as 1).
The Role of the Imaginary Unit
The imaginary unit, denoted by \(i\), is defined as \(i = \sqrt{-1}\). It provides a foundation for complex numbers, enabling computations that extend beyond the real number system.
In our exercise, the equation simplifies to \(z^2 = -1\). Solving such an equation is straightforward when we recognize the role of \(i\):
In our exercise, the equation simplifies to \(z^2 = -1\). Solving such an equation is straightforward when we recognize the role of \(i\):
- Taking the square root of both sides gives \(z = \pm i\).
- Thus, the fixed points are \(i\) and \(-i\).
Steps to Solve Equations for Fixed Points
Solving equations, especially when seeking fixed points, involves systematically manipulating the equation until you find where the input equals the output. Here鈥檚 a simple guide to tackle such equations:
- Set the function equal to its input: For fixed points, equate the function to the variable you are solving for (e.g., set \(w = z\)).
- Simplify the equation: Use techniques such as cross-multiplication to eliminate fractions.
- Rearrange and simplify further if needed: For instance, combine like terms.
- Solve the resulting equation: Apply algebraic methods and recognize patterns, such as perfect squares, that may involve \(i\).
