Complex Numbers and Functions

Descartes called $\sqrt{-1}$ an "imaginary" number as a jab in 1637. The term stuck. The number stayed. Today it lives in every MP3 file, every JPEG, every Wi-Fi packet. FFT - the Fast Fourier Transform - turns 3 seconds of audio into 132,300 complex numbers in a millisecond. Without complex analysis there is no digital world. Descartes laughed at $i$ - Spotify does not run without it.

• **MP3 and FLAC**: FFT decomposes an audio signal into frequencies via $e^{i\omega t}$. MP3 compression is a threshold on complex amplitudes - remove what the ear cannot hear.
• **JPEG and WebP**: the 2D discrete cosine transform is a direct descendant of the complex Fourier transform. Every photo is stored as a set of complex coefficients, not pixels.
• **Wi-Fi and 5G**: OFDM modulation operates with $N$ complex subcarriers simultaneously. Each data packet is an IFFT of a vector of complex symbols.
• **MRI scanners**: k-space in MRI is the direct Fourier domain. A brain image is obtained by inverse FFT of complex measurements.
• **AC circuits**: impedance $Z = R + i\omega L$ is a complex number. All AC circuit calculations are algebra over $\mathbb{C}$.

The Complex Plane

$x^2 + 1 = 0$ has no solutions in the real numbers. Descartes knew this and called $\sqrt{-1}$ "imaginary" - imaginaire - as a jab. 140 years passed. Fourier published his theory of heat conduction. Another 100 years and Fourier's theory was applied to radio signals. Another 50 - to digital audio. Today the "imaginary" number $i$ with $i^2 = -1$ is the foundation of every MP3 player. $z = a + bi$ is a **complex number**: $a$ is the real part Re($z$), $b$ is the imaginary part Im($z$).

A complex number z = a + bi is represented as a point on the **Argand plane** (the complex plane). The x-axis is the real part Re(z), the y-axis is the imaginary part Im(z). The modulus |z| = sqrt(a² + b²) is the distance from the origin. The conjugate z̄ = a - bi is the reflection across the real axis.

Arithmetic: addition is component-wise (like vectors), multiplication follows $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$. Division: $z/w = z\bar{w} / |w|^2$. Every nonzero complex number is invertible - $\mathbb{C}$ is algebraically closed. This means: every polynomial of degree $n$ over $\mathbb{C}$ has exactly $n$ roots. The reals cannot do this.

Polar Form

Instead of Cartesian coordinates $(a, b)$, a complex number is expressed through its **modulus** $r = |z|$ and **argument** $\theta = \arg(z)$ - the angle the vector $z$ makes with the positive Re axis. Then $z = r(\cos\theta + i\sin\theta)$. Polar form. This is where multiplication stops being a messy formula and becomes intuitive geometry.

The main advantage of polar form: **multiplication** becomes elementary. If z₁ = r₁·e^(iθ₁) and z₂ = r₂·e^(iθ₂), then z₁·z₂ = r₁r₂ · e^(i(θ₁+θ₂)). Moduli multiply, arguments add. Multiplication = scaling + rotation.

Exponentiation: $z^n = r^n e^{in\theta}$. De Moivre's formula: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$. The $n$-th roots of $z$ are $n$ points evenly spaced on a circle of radius $r^{1/n}$. Specifically, the $N$-th roots of unity $\omega_N^k = e^{2\pi ik/N}$ are the engine of the FFT algorithm.

The Complex Exponential and Euler's Formula

The expression $z = r(\cos\theta + i\sin\theta)$ compresses to $z = re^{i\theta}$ via Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$. Where does it come from? Taylor series. Substitute $x = i\theta$ into $e^x = 1 + x + x^2/2! + \ldots$ and separate real and imaginary parts - out comes exactly $\cos\theta + i\sin\theta$. Not magic - consequence of analysis.

Euler's formula: **e^(iθ) = cos θ + i sin θ**. Setting θ = π gives **e^(iπ) + 1 = 0** - Euler's identity, linking five fundamental constants: e, i, π, 1, 0. Richard Feynman called it "the most remarkable formula in mathematics".

The deep meaning: **the exponential is rotation**. Motion along the imaginary axis ($i\theta$) becomes rotation around the unit circle. Not a coincidence - the exponential is the solution to $f'(t) = if(t)$, which describes uniform rotation at angular velocity 1. A Wi-Fi receiver does exactly this: it synchronizes its local oscillator with the carrier by multiplying by $e^{-i\omega t}$.

Now the polar form is written compactly: **z = r·e^(iθ)**. Multiplication: z₁·z₂ = r₁r₂·e^(i(θ₁+θ₂)). Division: z₁/z₂ = (r₁/r₂)·e^(i(θ₁-θ₂)). Power: z^n = r^n·e^(inθ). Everything works like ordinary exponential arithmetic.

Elementary Functions of a Complex Variable

Take familiar functions - exp, sin, cos, log - and extend them to complex numbers. Something breaks. Or rather, opens up. $\sin(z)$ is no longer bounded to $[-1, 1]$. The exponential becomes periodic. The logarithm splits into infinitely many values. Each of these 'violations' is not a defect but a new tool.

The complex exponential: e^z = e^(a+bi) = e^a · e^(bi) = e^a(cos b + i sin b). It is **periodic** with period 2πi: e^(z+2πi) = e^z. In real analysis the exponential is monotone, but in the complex realm it is periodic!

The most striking fact: the complex **logarithm is multivalued**. $\log z = \ln|z| + i(\arg(z) + 2\pi k)$ for any integer $k$. Choosing $k = 0$ and $-\pi < \arg(z) \le \pi$ gives the **principal value** $\text{Log}(z)$. Multivaluedness leads to the concept of a Riemann surface - a geometric object that 'unrolls' all branches of the logarithm into a single sheet. Not exotic: the same multivaluedness appears when analyzing impedance in AC circuits and computing transfer functions.

Back to the beginning: Descartes called $\sqrt{-1}$ "imaginary" as a jab. Complex numbers turned out to be not a trick but the language of physics. Functions behave more richly, polynomials always have roots (fundamental theorem of algebra), and Euler's identity $e^{i\pi} + 1 = 0$ unites five fundamental constants in one equation. Richard Feynman called it "the most remarkable formula in mathematics".

Key Ideas

• **Complex number** $z = a + bi$ - a point on the Argand plane: modulus $|z| = \sqrt{a^2 + b^2}$, argument $\arg(z)$ - the angle with the Re axis
• **Polar form** $z = re^{i\theta}$: multiplication = rotation + scaling. This is how OFDM in Wi-Fi works - each symbol is a rotation of a vector in $\mathbb{C}$
• **Euler's formula** $e^{i\theta} = \cos\theta + i\sin\theta$ - not a pretty trick but a consequence of Taylor series. It turns FFT from $O(N^2)$ to $O(N \log N)$
• **Complex functions** behave more richly: $\sin$ is not bounded to $[-1, 1]$, the exponential is periodic with $T = 2\pi i$, the logarithm is multivalued - these are not bugs but features from which residue theory is built
• **Callback**: Descartes laughed at $i$ in 1637 - every signal processing algorithm in the world runs on it today

Related Topics

Complex numbers are the foundation of complex analysis:

• Analytic Functions — Differentiability in the complex sense is far stronger than in the real sense
• Complex Integration — Contour integrals - a powerful tool built on complex structure

Вопросы для размышления

• Why is the complex exponential periodic ($e^{z+2\pi i} = e^z$) while the real one is monotone? What happens geometrically when moving along the imaginary axis?
• FFT runs in $O(N \log N)$ instead of $O(N^2)$ - what role do the $N$-th roots of unity $\omega_N^k = e^{2\pi i k/N}$ play in that speedup?
• Quaternions $\mathbb{H}$ extend $\mathbb{C}$ to 4 dimensions. Where are they used in game engines, and why is 3D rotation better described by quaternions than by Euler angle matrices?

Связанные уроки

• calc-16-taylor