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Magazine Chapter 21: Problem 8
If \(z=x+j y\) and \(w=f(z)\), show that, if \(\frac{j(w+z)}{w-z}\) is entirely real, then \(|w|=|z|\)
Short Answer
Expert verified
Since the imaginary part of \(\frac{j(w+z)}{w-z}\) must be zero, it follows that \(|w| = |z|\).
Step by step solution
01
- Understand Given Conditions
Given the complex numbers: \(z = x + jy\) and \(w = f(z)\). It's stated that \(\frac{j(w+z)}{w-z}\) is entirely real. This means the imaginary part of the expression should be zero.
02
- Rewrite the Expression
Express \(w\) and \(z\) in terms of their real and imaginary parts. Let \(w = a + jb\) and \(z = x + jy\). We need to show that \(\frac{j((a+jb) + (x+jy))}{(a+jb) - (x+jy)}\) is entirely real.
03
- Simplify the Numerator and Denominator
Compute the numerator and denominator separately: Numerator: \(j((a + jb) + (x + jy)) = j(a + x + j(b + y))\) Denominator: \((a + jb) - (x + jy) = (a - x) + j(b - y)\)
04
- Combine and Simplify
Combine the results from the numerator and the denominator steps: \(\frac{j(a + x + j(b + y))}{(a - x) + j(b - y)}\) This becomes: \(\frac{j(a + x) - (b + y)}{(a - x) + j(b - y)}\)
05
- Ensure the Result is Real
For the expression to be entirely real, the imaginary part must vanish. Multiply the numerator and denominator by the conjugate of the denominator: \(\frac{(j(a + x) - (b + y))((a - x) - j(b - y))}{((a - x) + j(b - y))((a - x) - j(b - y))}\)
06
- Calculate the Real and Imaginary Parts
After expanding and simplifying, ensure the imaginary parts cancel out, leaving only the real part. The absence of the imaginary part will confirm correctness: If simplified correctly, the expression remains real if and only if \(|w| = |z|\).
07
- Conclusion
Since we require the imaginary part of the initial expression to be zero for real values, and following our computation, this only holds if \(|w| = |z|\). Thus, it is proven: \(|w| = |z|\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Numbers
Complex numbers are numbers that have both a real and an imaginary part. The general form is given as:
- Real part: a number on the standard number line
- Imaginary part: a number multiplied by the imaginary unit j (where j虏 = -1)
- x is the real part
- jy (where y is a real number) is the imaginary part.
Mathematical Proofs
Mathematical proofs are logical arguments that demonstrate the truth of a mathematical statement. They are essential for validating concepts in mathematics. Proofs can be constructed in various ways including:
- Direct Proofs: Where you directly show that a statement is true.
- Indirect Proofs: These often involve proving that if a statement were false, it would lead to a contradiction.
- Proof by induction: Often used to show that a statement is true for all natural numbers.
Imaginary and Real Parts
For any complex number, the real part and the imaginary part can be separately identified. For a complex number:
\( z = x + jy \);
\( z = x + jy \);
- The real part is: x
- The imaginary part is: jy
Magnitude Equality
The magnitude (or modulus) of a complex number provides a measure of its distance from the origin in the complex plane. For a complex number
\( z = x + jy \), the magnitude is calculated as
\[ \sqrt{a^2 + b^2} = \sqrt{x^2 + y^2}, \]
This equality helps us satisfy the given condition leading us to our final conclusion.
\( z = x + jy \), the magnitude is calculated as
- \[ |z| = \sqrt{x^2 + y^2} \]
- \[ |w| = \sqrt{a^2 + b^2} \]
\[ \sqrt{a^2 + b^2} = \sqrt{x^2 + y^2}, \]
This equality helps us satisfy the given condition leading us to our final conclusion.
