Euler's Formula

Deep Breakdown

What it tells me (Situation Awareness): Euler's formula establishes the deep connection between the complex exponential and trigonometric functions. It says that raising $e$ to a purely imaginary exponent $ix$ produces a unit complex number whose real part is $\cos x$ and imaginary part is $\sin x$. This is arguably the most important formula connecting analysis, trigonometry, and complex numbers. In DA/DS and ML: Euler's formula is the basis of all Fourier analysis (FFT), signal processing, and any system where exponential growth and oscillation interact.

Flowchart

Real-world use

Where it is used

Fourier series and DFT: basis functions are $e^{i2\pi kn/N} = \cos(2\pi kn/N) + i\sin(2\pi kn/N)$. These are exactly Euler's formula evaluated at $x = 2\pi kn/N$. Signal modulation: multiplying a signal by $e^{i2\pi f_0 t}$ shifts its frequency by $f_0$. Solving ODEs with oscillatory solutions: solutions of the form $e^{(\alpha + i\omega)t} = e^{\alpha t}(\cos\omega t + i\sin\omega t)$.

Problem it solves

The DFT of a signal decomposes it into $e^{i2\pi kn/N}$ basis functions. Without Euler's formula, these complex exponentials would be mysterious. With it: they are just sine and cosine waves combined into one compact notation. Every FFT coefficient $X[k]$ measures how much of the frequency $k$ basis wave $(\cos + i\sin)$ is present in the signal.

How it solves the problem

$e^{ix} = \cos x + i\sin x$ means every point on the unit circle (modulus 1, angle $x$) can be written as $e^{ix}$. The exponential function unifies linear growth (real exponent) with oscillation (imaginary exponent). Complex exponentials $e^{(a+bi)t} = e^{at}(\cos bt + i\sin bt)$ model systems with simultaneous growth/decay and oscillation.

Example in a business

Time series Fourier decomposition: the DFT coefficient $X[k] = \sum_{n=0}^{N-1} x[n] e^{-i2\pi kn/N}$. By Euler: $e^{-i2\pi kn/N} = \cos(2\pi kn/N) - i\sin(2\pi kn/N)$. The real and imaginary parts of $X[k]$ measure the cosine and sine components at frequency $k/N$ cycles per sample. This is how seasonality is detected: monthly, quarterly, annual cycles each correspond to a specific $k$.

The formula, piece by piece

Breaking the notation:

• $e$: Euler's number ($\approx 2.71828$), the base of the natural exponential.
• $i$: the imaginary unit ($i^2 = -1$).
• $x$: a real number (the angle in radians).
• $e^{ix}$: the complex exponential — Euler's formula is its definition via Taylor series.
• $\cos x$: real part of $e^{ix}$.
• $i\sin x$: imaginary part of $e^{ix}$.

Derivation via Taylor series (motivational): $e^{ix} = \sum_{n=0}^{\infty} \frac{(ix)^n}{n!} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \cdots$ $= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\right)$ $= \cos x + i\sin x$.

Key consequences:

• $e^{-ix} = \cos x - i\sin x$ (conjugate).
• $\cos x = \frac{e^{ix} + e^{-ix}}{2}$.
• $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$.
• $|e^{ix}| = 1$ for all real $x$ (unit circle).
• DFT: $X[k] = \sum_n x[n] e^{-i2\pi kn/N}$.

The formula in words

"The complex exponential $e^{ix}$ is a unit-length 'arrow' in the complex plane pointing at angle $x$ radians. Its horizontal shadow is $\cos x$ and its vertical shadow is $\sin x$. As $x$ increases from 0 to $2\pi$, the arrow sweeps a complete circle. This is why frequencies are represented as complex exponentials: they describe circular motion, which produces sinusoidal oscillations."

Critical Concepts

1. DFT basis: every DFT is a dot product of the signal with $e^{-i2\pi kn/N}$ (a complex exponential via Euler). np.fft.fft internally uses these basis functions.
2. Frequency shift theorem: multiplying a signal by $e^{i2\pi f_0 t}$ shifts its frequency spectrum by $f_0$. This is the basis of modulation (AM/FM radio, OFDM in 5G).
3. $e^{ix}$ is periodic: $e^{i(x + 2\pi)} = e^{ix}$. This periodicity is why the DFT is periodic with period $N$.
4. Complex exponential vs. real exponential: $e^{ax}$ for real $a > 0$ grows without bound; $e^{ix}$ for real $x$ stays on the unit circle.

They combine in $e^{(a+ix)} = e^a \cdot e^{ix}$: growth rate $e^a$ and oscillation $e^{ix}$ simultaneously.

Step-by-step calculation (tiny example)

Task: verify Euler's formula at $x = \pi/2$ and $x = \pi$.

Step 1: $x = \pi/2$ $e^{i\pi/2} = \cos(\pi/2) + i\sin(\pi/2) = 0 + i \cdot 1 = i$. So $e^{i\pi/2} = i$: a quarter turn counter-clockwise gives the imaginary unit.

Step 2: $x = \pi$ $e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1 + i \cdot 0 = -1$. This gives Euler's identity: $e^{i\pi} + 1 = 0$.

Step 3: $x = 2\pi$ $e^{i2\pi} = \cos(2\pi) + i\sin(2\pi) = 1 + 0i = 1$. A full rotation returns to the starting point.

Code examples

Beginner example (Python)

# Euler's Formula: e^(ix) = cos(x) + i*sin(x)
import numpy as np

angles = [0, np.pi/6, np.pi/4, np.pi/2, np.pi, 3*np.pi/2, 2*np.pi]
labels = ["0", "π/6", "π/4", "π/2", "π", "3π/2", "2π"]
print("Euler's Formula: e^(ix) = cos(x) + i·sin(x)")
print(f"{'x':>6} {'e^(ix)':>18} {'cos+i·sin':>18} {'|e^(ix)|':>10}")
for x, label in zip(angles, labels):
    euler = np.exp(1j * x)              # e^(ix) via NumPy
    trig = np.cos(x) + 1j * np.sin(x)   # cos(x) + i*sin(x)
    modulus = abs(euler)
    equal = abs(euler - trig) < 1e-10
    print(f" {label:>4} {str(euler.round(4)):>18} {str(trig.round(4)):>18} {modulus:.4f} ✓" if equal else "MISMATCH")

# DFT basis functions via Euler's formula
N = 8
print(f"\nDFT basis functions e^(-i·2π·k·n/N) for N={N}, n=0..7 and k=0,1,2:")
for k in range(3):
    basis = np.array([np.exp(-1j * 2 * np.pi * k * n / N) for n in range(N)])
    print(f" k={k}: {np.round(basis, 3)}")

Beginner example (R)

# Euler's Formula in R: exp(1i * x)
angles <- c(0, pi/6, pi/4, pi/2, pi, 3*pi/2, 2*pi)
labels <- c("0", "π/6", "π/4", "π/2", "π", "3π/2", "2π")
cat("Euler's Formula: e^(ix) = cos(x) + i·sin(x):\n")
for (idx in seq_along(angles)) {
  x <- angles[idx]
  euln <- exp(1i * x)
  trig <- cos(x) + 1i * sin(x)
  cat(sprintf(" x=%-5s e^(ix)=%.4f+%.4fi |e^(ix)|=%.4f\n", labels[idx], Re(euln), Im(euln), Mod(euln)))
}

Professional example (Python)

import numpy as np
import pandas as pd

# Scenario: Fourier decomposition of a sales time series
# FFT coefficients ARE Euler's formula applied at discrete frequencies
np.random.seed(42)
N = 120  # 10 years monthly
t_months = np.arange(N)
sales = (
    50_000 + 500 * t_months                          # linear trend
    + 10_000 * np.cos(2 * np.pi * t_months / 12)     # annual cycle
    + 3_000 * np.cos(2 * np.pi * t_months / 3)       # quarterly cycle
    + np.random.normal(0, 1_500, N)                  # noise
)

# DFT via FFT: X[k] = sum_n x[n] * e^(-i*2π*kn/N)
fft_raw = np.fft.rfft(sales)
freqs = np.fft.rfftfreq(N, d=1)  # cycles per month

# Each coefficient is: Re[X[k]] = cos component, Im[X[k]] = sin component
amplitudes = np.abs(fft_raw)
phases = np.angle(fft_raw)
print("Fourier Decomposition via Euler's Formula (e^(ix) = cos(x) + i·sin(x))")
print("=" * 70)
print(f" {'k':>3} {'freq (cpm)':>12} {'period (mo)':>12} {'amplitude':>12} {'phase (deg)':>12}")

# Show components for meaningful frequencies
for k in range(1, len(fft_raw)):
    f = freqs[k]
    if f == 0:
        continue
    period = 1 / f
    amp = amplitudes[k]
    phase = np.degrees(phases[k])
    if amp > 2000:  # only significant components
        print(f" {k:>3} {f:>12.4f} {period:>12.1f} {amp:>12.1f} {phase:>12.1f}")

print()
print(" Euler's formula makes this explicit:")
print(" X[k] = Σ x[n]·(cos(2πkn/N) - i·sin(2πkn/N))")
print(" = Σ x[n]·cos(...) - i·Σ x[n]·sin(...)")
print(" Re[X[k]] = cosine projection = how much 'cosine at freq k' is in signal")
print(" Im[X[k]] = sine projection = how much 'sine at freq k' is in signal")
print()

# Demonstrate: manual DFT for k=10 (1/12 months = annual cycle)
k_annual = round(N / 12)
basis_cos = np.cos(2 * np.pi * k_annual * t_months / N)
basis_sin = np.sin(2 * np.pi * k_annual * t_months / N)
X_re_manual = np.sum(sales * basis_cos)
X_im_manual = np.sum(sales * (-basis_sin))
X_fft = fft_raw[k_annual]
print(f"Manual DFT at annual frequency (k={k_annual}):")
print(f" Re[X[{k_annual}]] manual = {X_re_manual:.1f} | fft = {X_fft.real:.1f}")
print(f" Im[X[{k_annual}]] manual = {X_im_manual:.1f} | fft = {X_fft.imag:.1f}")
print(f" Amplitude = |X[{k_annual}]| = {abs(X_fft):.1f}")

Professional example (R)

# Manual DFT via Euler's formula: Re = sum(x*cos), Im = -sum(x*sin)
set.seed(42)
N <- 60
t_months <- 0:(N - 1)
sales <- 10000 + 500 * t_months + 2000 * sin(2 * pi * t_months / 12) + rnorm(N, 0, 200)
fft_raw <- fft(sales)
k_annual <- 5
basis_cos <- cos(2 * pi * k_annual * t_months / N)
basis_sin <- sin(2 * pi * k_annual * t_months / N)
X_re_manual <- sum(sales * basis_cos)
X_im_manual <- -sum(sales * basis_sin)
X_fft <- fft_raw[k_annual + 1]
cat(sprintf("Manual DFT at k=%d: Re = %.1f (fft %.1f), Im = %.1f (fft %.1f)\n", k_annual, X_re_manual, Re(X_fft), X_im_manual, Im(X_fft)))
cat(" Amplitude = |X| =", round(Mod(X_fft), 1), "\n")

Source: domain knowledge

#foundations#number-sets-and-logic#complex-numbers-foundations