Question

This problem is to refresh your memory about some properties of complex numbers needed at several points in this chapter, but especially in deriving the resonance formula (5.64). (a) Prove that any complex number \(z=x+i y\) (with \(x\) and \(y\) real) can be written as \(z=r e^{i \theta}\) where \(r\) and \(\theta\) are the polar coordinates of \(z\) in the complex plane. (Remember Euler's formula.) (b) Prove that the absolute value of \(z\), defined as \(|z|=r,\) is also given by \(|z|^{2}=z z^{*},\) where \(z^{*}\) denotes the complex conjugate of \(z\) defined as \(z^{*}=x-\)iy. \(\left(\text { c) Prove that } z^{*}=r e^{-i \theta} . \text { (d) Prove that }(z w)^{*}=z^{*} w^{*} \text { and that }(1 / z)^{*}=1 / z^{*}\right.\) (e) Deduce that if \(z=a /(b+i c),\) with \(a, b,\) and \(c\) real, then \(|z|^{2}=a^{2} /\left(b^{2}+c^{2}\right)\).

Short answer

The complex number in polar form is \(z = re^{i\theta}\). Absolute value squared \(|z|^2 = z z^*\), and \(z^* = re^{-i\theta}\). With \(z = a/(b+ic)\), \(|z|^2 = a^2/(b^2+c^2)\).

Step-by-step solution

Understand the Complex Number in Polar Form

A complex number can be expressed in two forms: Cartesian form \(z = x + iy\) and polar form \(z = r e^{i \theta}\), where \(r\) is the magnitude \(|z|\) and \(\theta\) is the angle (argument) with the positive x-axis. By Euler's formula, \(e^{i \theta} = \cos \theta + i \sin \theta\), thus any complex number can be written as \(z = r (\cos \theta + i \sin \theta)\).

Use Euler's Formula

Rewriting using Euler's formula, \(z = r e^{i \theta}\), where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\). This verifies the representation \(z = re^{i\theta}\).

Prove Absolute Value and Conjugate Properties

The absolute value \(|z|\) is \(|z| = r\). The squared absolute value is given by \(|z|^2 = z \cdot z^{*}\), where \(z^{*}\) is the complex conjugate of \(z\), \(z^{*} = x - iy\). So, \(zz^* = (x+iy)(x-iy) = x^2 + y^2 = r^2\).

Prove Polar Conjugate Form

Express \(z^{*}\) in polar form: from \(z = re^{i\theta}\), the conjugate \(z^* = re^{-i\theta}\), because the complex conjugate involves negating the imaginary part anglex. Thus \(r (\cos(-\theta) + i\sin(-\theta)) = r(\cos \theta - i\sin \theta)\).

Multiplicative Conjugate Property

Consider the product \(w\) and its conjugate. We write \((zw)^* = (z w)^* = z^* w^*\), showing conjugation distributes over multiplication.

Division Conjugate Property

For \(\frac{1}{z}\), the conjugate is \((\frac{1}{z})^* = \frac{1}{z^*}\), since dividing by a complex number is equivalent to multiplying by its conjugate over its magnitude squared.

Solve for Polar Conjugate Simplification

Given \(z = \frac{a}{b+ic}\), multiply numerator and denominator by the conjugate of the denominator: \(z = \frac{a(b-ic)}{(b+ic)(b-ic)} = \frac{ab - aic}{b^2 - (ic)^2} = \frac{ab - aic}{b^2 + c^2}\).

Final Check and Form Verification

Check that \(|z|^2 = a^2 / (b^2 + c^2)\) follows from this representation by showing the logic holds: multiplying the denominator and numerator by the conjugate is shown to equate to the absolute value squared, providing final proof.

Key concepts

Polar Coordinates

Polar coordinates are a way to express complex numbers using a radius and an angle. This system is particularly useful in mathematics and physics where the position of a point is determined by how far away it is from a reference point, referred to as the origin, and the angle from a reference direction. In the complex plane, any complex number \( z = x + iy \) can be written as \( z = r e^{i \theta} \), where:

• \( r \) is the absolute value or magnitude of \( z \), calculated as \( r = \sqrt{x^2 + y^2} \).
• \( \theta \) is the argument, representing the angle \( z \) makes with the positive x-axis, given by \( \theta = \tan^{-1}(y/x) \).

Understanding the polar coordinates allows us to visualize complex numbers in a geometric way, enabling easier calculations and transformations.

Euler's Formula

Euler's formula is a fundamental bridge between trigonometry and complex numbers. It states that for any real number \( \theta \), \( e^{i \theta} = \cos \theta + i \sin \theta \). This formula is incredibly powerful:

• It allows us to convert between exponential forms and trigonometric forms of complex numbers, creating a compact representation in the complex plane.
• It simplifies the manipulation of complex number expressions, especially when multiplying and dividing complex numbers.

This is why any complex number can be expressed as \( z = r e^{i \theta} \) using Euler's formula, where \( r \) and \( \theta \) are its polar coordinates.

Complex Conjugate

The complex conjugate of a complex number is a reflection over the real axis in the complex plane. For a given complex number \( z = x + iy \), the complex conjugate is noted as \( z^* = x - iy \). This operation is key for:

• Solving and simplifying complex equations.
• Finding the magnitude of complex numbers since \( |z|^2 = z z^* \), which equals \( x^2 + y^2 \).
• Balancing denominators in division by complex numbers by multiplying by the conjugate, eliminating imaginary components.

In polar form, if \( z = re^{i\theta} \), then \( z^* = re^{-i\theta} \), leveraging Euler's formula, to negate the imaginary part.

Absolute Value

The absolute value of a complex number is its distance from the origin in the complex plane. For a complex number \( z = x + iy \), its absolute value is calculated as \( |z| = \sqrt{x^2 + y^2} \). Key points about absolute value include:

• In polar form, it is simply the radius \( r \).
• The square of the absolute value is given by \( |z|^2 = z z^* \), demonstrating a neat relationship with the complex conjugate.
• It provides the modulus necessary for calculating distance and magnitude in many fields, including physics and engineering.

The absolute value always outputs a non-negative real number, which helps in comparing the size of complex numbers.

Polar Form

The polar form of expressing complex numbers combines magnitude and angle information into a single formula. When a complex number is expressed as \( z = r e^{i\theta} \), it is in polar form where \( r \) denotes magnitude and \( \theta \) is the angle. Benefits of polar form include:

• It simplifies multiplication and division of complex numbers, as magnitudes are multiplied and angles are added or subtracted.
• It provides an intuitive geometric interpretation which makes graphing complex numbers straightforward.
• It helps in efficiently solving equations involving powers and roots of complex numbers due to ease in handling exponents.

This form ties back to Euler's formula and showcases the elegance of complex number algebra.

Resonance Formula

The resonance formula often arises in physics when dealing with oscillating systems like electrical circuits or mechanical systems. It involves complex numbers frequently, especially in expressing impedance or response functions. When represented in complex form, tasks like finding resonance conditions can become simpler.

• Complex numbers allow encapsulation of amplitude and phase of oscillations, which are crucial in resonance analysis.
• Using polar coordinates simplifies calculations, ensuring clarity in the phase angles involved.
• Euler's and other formulas incorporating complex conjugates demonstrate shifts in resonance frequencies efficiently.

Understanding and working with resonance often involves leveraging complex conjugates and absolute values to assess and optimize system responses at various frequencies.