Mathematics for Physics I - Complex Numbers

Definition and Representation

Complex numbers extend the real number system by introducing the imaginary unit \(i\), where \(i^2 = -1\). A complex number \(z\) can be written in the form:

\[ z = a + bi \]

where \(a\) and \(b\) are real numbers, \(a\) is the real part, and \(b\) is the imaginary part

Complex numbers can be represented in several ways:

Rectangular Form

\(z = a + bi\)

Where \(a = \text{Re}(z)\) and \(b = \text{Im}(z)\)

Polar Form

\(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\)

Where \(r = |z| = \sqrt{a^2 + b^2}\) and \(\theta = \arg(z) = \tan^{-1}(b/a)\)

Interactive visualization of complex numbers in the complex plane

Operations with Complex Numbers

Addition and Subtraction

If \(z_1 = a + bi\) and \(z_2 = c + di\), then:

\[ z_1 + z_2 = (a + c) + (b + d)i \] \[ z_1 - z_2 = (a - c) + (b - d)i \]

Addition and subtraction are performed component-wise, just like vector addition.

Multiplication

If \(z_1 = a + bi\) and \(z_2 = c + di\), then:

\[ z_1 \cdot z_2 = (ac - bd) + (ad + bc)i \]

In polar form, multiplication is particularly elegant:

\[ r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1 + \theta_2)} \]

This shows that multiplying complex numbers multiplies their magnitudes and adds their angles.

Division

If \(z_1 = a + bi\) and \(z_2 = c + di\), then:

\[ \frac{z_1}{z_2} = \frac{(a + bi)}{(c + di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]

In polar form, division is:

\[ \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)} \]

Complex Conjugate

The complex conjugate of \(z = a + bi\) is:

\[ \bar{z} = a - bi \]

Properties of complex conjugates:

\(z \cdot \bar{z} = |z|^2 = a^2 + b^2\)
\(\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2\)
\(\overline{z_1 \cdot z_2} = \bar{z}_1 \cdot \bar{z}_2\)

Euler's Formula and De Moivre's Theorem

Euler's Formula

\[ e^{i\theta} = \cos\theta + i\sin\theta \]

This remarkable formula connects complex exponentials to trigonometric functions and is fundamental in physics and engineering.

De Moivre's Theorem

\[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \]

Or using Euler's formula:

\[ (e^{i\theta})^n = e^{in\theta} \]

This theorem is particularly useful for finding powers and roots of complex numbers.

Visualization of Euler's formula showing the relationship between complex exponentials and the unit circle

Applications in Physics

Oscillatory Motion

Complex numbers provide a compact way to represent oscillations:

\[ x(t) = A\cos(\omega t + \phi) = \text{Re}(Ae^{i(\omega t + \phi)}) \]

This representation simplifies the mathematics of harmonic oscillators, waves, and AC circuits.

AC Circuit Analysis

Impedance in AC circuits is represented as a complex number:

\[ Z = R + iX \]

Where \(R\) is resistance and \(X\) is reactance. This allows for elegant analysis of circuits with capacitors and inductors.

Quantum Mechanics

The Schrödinger equation and wave functions are inherently complex:

\[ i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi \]

Complex numbers are essential for describing quantum states and their evolution.

Signal Processing

The Fourier transform uses complex exponentials to decompose signals:

\[ F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}dt \]

This is fundamental in analyzing waves, signals, and fields in physics.

Practice Problems

Test your understanding of complex numbers with these practice problems:

Express the complex number \(z = 3 - 4i\) in polar form.
Find all the cube roots of \(z = 8i\).
If \(z_1 = 2 + 3i\) and \(z_2 = 1 - i\), calculate \(z_1 \cdot z_2\) and \(\frac{z_1}{z_2}\).
Prove that \(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\) for any complex numbers \(z_1\) and \(z_2\).
Use Euler's formula to derive the identity \(\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)\).

Key Insight:

Complex numbers are not just mathematical abstractions—they provide powerful tools for solving real-world problems in physics and engineering. Their geometric interpretation as points in the complex plane helps visualize operations that would otherwise be difficult to conceptualize.