Euler's Formula — The Circle as an Exponential

1. History — Euler in 1748, and the long road to i

Leonhard Euler published the formula in its modern form in Chapter VIII of Introductio in analysin infinitorum (1748). He wrote, with his characteristic clarity, that for any real φ, e^+iφ = cos φ + i sin φ and e^−iφ = cos φ − i sin φ. From these, the cosine and sine followed as the even and odd parts of the exponential.

Euler was not working in a vacuum. Complex numbers had been regarded with suspicion since Cardano's Ars Magna (1545). Descartes called them "imaginary" in 1637, intending dismissal. Yet by the early 1700s, they proved unavoidable in solving cubics and in integration.

Roger Cotes, in 1714, had essentially discovered the logarithmic form: iφ = ln(cos φ + i sin φ). He died at 33, and his Harmonia Mensurarum (1722) was published posthumously. The insight remained obscure.

Why 1748 mattered

Infinite series as definition. Euler treated e^z, sin z, cos z as power series, allowing z to be complex without hand-waving.
Notation standardization. Euler popularized e, i, π, f(x), Σ — the language that lets the formula be read at a glance.
Analytic continuation. The exponential, defined for reals by limits, was extended to the complex plane by its series.

Euler was already losing his sight. He would become fully blind in 1771, yet produce nearly half his work thereafter — dictating formulas that he visualized with extraordinary internal clarity.

Abraham de Moivre in 1707 and 1722 proved (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for integer n, by induction on trigonometry. He did not write it with e. Johann Bernoulli had manipulated e^ix in correspondence with Leibniz around 1702, but with confusion about logarithms of negatives.

Euler's synthesis was to recognize that the exponential — the eigenfunction of d/dx — when fed an imaginary argument, must rotate rather than grow. The Taylor series makes this mechanically obvious: even powers of i are real, odd powers are imaginary, and signs alternate exactly as in cosine and sine.

The cultural resistance dissolved slowly. Wessel (1799), Argand (1806), and Gauss (1831) gave the complex plane a geometric home. Once you could see multiplication by i as a 90° rotation, Euler's formula ceased to be magic and became definition: e^iθ is the point you reach after walking θ radians around the unit circle.

Insight: Before Euler, trigonometry and exponentials lived in separate books. After Euler, they are two views of one object — circular motion is exponential motion with imaginary time.

2. Three Derivations — series, differential equation, limit

2.1 Taylor series (the classic proof)

Define the exponential for all complex z by its absolutely convergent series:

e^z = 1 + z + z²/2! + z³/3! + z⁴/4! + …

Set z = iθ with θ real. Use i² = −1, i³ = −i, i⁴ = 1, repeating every four terms:

e^{iθ} = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + iθ⁵/5! − θ⁶/6! − …
      = (1 − θ²/2! + θ⁴/4! − θ⁶/6! + …) + i(θ − θ³/3! + θ⁵/5! − …)

The parentheses are precisely the Taylor series for cos θ and sin θ. Hence e^iθ = cos θ + i sin θ. No geometric assumption is needed beyond the series definitions; convergence for all θ follows from ratio test.

2.2 Differential equation (dynamical proof)

Let f(θ) = cos θ + i sin θ. Differentiate: f'(θ) = −sin θ + i cos θ = i(cos θ + i sin θ) = i f(θ), and f(0)=1. The unique solution to f' = i f with f(0)=1 is the exponential e^iθ, because the exponential is characterized by being its own derivative. Formally, consider g(θ)=f(θ)e^−iθ. Then g' = f' e^−iθ − i f e^−iθ=0, so g is constant =g(0)=1. Thus f = e^{iθ}.

This viewpoint explains why the formula feels inevitable: both sides describe the same uniform circular motion — unit speed, starting at 1, always turning left at 90° to the radius.

2.3 Limit definition (geometric multiplication)

Recall e^{x}=lim_{n→∞}(1+x/n)^{n}. For imaginary x=iθ:

e^{iθ} = lim (1 + iθ/n)^{n}

Each factor (1 + iθ/n) is a complex number slightly above 1 on the real axis: magnitude √(1+(θ/n)²)≈1, argument arctan(θ/n)≈θ/n. Multiplying n copies adds the arguments: total angle ≈ n·θ/n = θ, while the magnitude →1. Geometrically you walk n tiny steps, each turning θ/n left; the polygon converges to an arc of the unit circle.

All three derivations agree because the complex exponential is entire — the same function arises from series, ODE, and limit.

3. Geometry — the unit circle in the complex plane

Plot a complex number z = x + iy as the point (x,y). Its modulus |z| = √(x²+y²) is distance from origin; its argument arg z = θ is the angle from the positive real axis. Euler's formula gives the polar form:

z = r e^{iθ} where r=|z|, θ=arg z

Multiplication becomes beautifully simple: r₁e^{iθ₁} · r₂e^{iθ₂} = r₁r₂ e^{i(θ₁+θ₂)}. Magnitudes multiply, angles add. In particular, multiplication by e^{iθ} is a pure rotation by θ about the origin.

θ	e^{iθ}	Geometric meaning
0	1	identity
π/2	i	90° CCW
π	−1	half-turn
3π/2	−i	270° CCW
2π	1	full turn

The map θ ↦ e^{iθ} is a group homomorphism from (ℝ,+) to the unit circle (S¹,×). It wraps the infinite real line onto the circle, with period 2π. Its derivative is i e^{iθ}, a vector of length 1 tangent to the circle, confirming unit speed parametrization: arc length from 0 to θ is exactly θ.

This is why engineers write oscillations as Re[A e^{iωt}]. The real part projects the uniform circular motion onto the horizontal axis, yielding a cosine; the imaginary part yields sine.

4. Consequences — identity, de Moivre, roots of unity

Euler's identity

Set θ=π: cos π = −1, sin π = 0. Then e^{iπ} + 1 = 0. Five fundamental constants — e, i, π, 1, 0 — linked by three operations. It is not mystical; it is the statement that a half-turn around the unit circle lands at −1.

de Moivre's formula

From (e^{iθ})^{n}=e^{inθ} we obtain (cos θ + i sin θ)^{n}=cos nθ + i sin nθ for integer n, and by continuity for real n. Expand with binomial theorem to derive multiple-angle identities without memorizing triangles:

cos 2θ = cos²θ − sin²θ, sin 2θ = 2 sinθ cosθ

Roots of unity

Solve z^{n}=1: write z = e^{iθ}. Then e^{inθ}=1 ⇒ nθ=2πk ⇒

z_k = e^{i2πk/n}, k=0,…,n−1

These n points form a regular n-gon. For n=3: 1, e^{i2π/3}=−1/2+i√3/2, e^{i4π/3}=−1/2−i√3/2. They are the vertices of an equilateral triangle and the basis of the discrete Fourier transform.

Complex trig and logs

Inverting Euler: cos θ = (e^{iθ}+e^{−iθ})/2, sin θ = (e^{iθ}−e^{−iθ})/(2i). Allow θ complex and you define cosine and sine for complex arguments. The complex logarithm becomes multi-valued: ln(re^{iθ}) = ln r + i(θ+2πk).

5. Applications — why the formula runs the modern world

Fourier analysis

Any periodic signal can be written as a sum of complex exponentials: f(t)=Σ c_n e^{inωt}. Euler's formula converts this to sines and cosines, but exponentials are eigenfunctions of linear time-invariant systems: differentiation becomes multiplication by inω, convolution becomes multiplication. The Fourier transform kernel e^{−i2πft} is literally Euler's formula in the integrand.

Electrical engineering — phasors

A sinusoid V(t)=V₀cos(ωt+φ) is represented as Re[ V̰ e^{iωt} ] with phasor V̰=V₀e^{iφ}. Inductor voltage L dI/dt becomes iωL Ḭ, capacitor becomes 1/(iωC). Differential equations reduce to complex algebra. Three-phase power, impedance matching, and phasor measurement units (PMUs) all rely on rotating vectors = e^{iθ}.

Quantum mechanics

Time evolution is unitary: |ψ(t)⟩ = e^{−iĤt/ħ}|ψ(0)⟩. A free particle is a plane wave e^{i(kx−ωt)} = cos(kx−ωt)+i sin(kx−ωt). Interference patterns arise from addition of phases. The Schrödinger equation's i forces oscillation rather than exponential blow-up, preserving probability.

Rotations and computer graphics

In 2D, rotate vector (x,y) by θ via multiplication by e^{iθ}. In matrix form this is [[cos,−sin],[sin,cos]]. In 3D, unit quaternions q = e^{uθ/2}=cos(θ/2)+u sin(θ/2) (with u a pure imaginary unit) generalize Euler's formula, avoiding gimbal lock in robotics and games.

Signal processing and control

The z-transform evaluated on the unit circle z=e^{iω} gives the discrete-time Fourier transform. Stability of digital filters depends on poles inside |z|<1. The Nyquist plot maps the imaginary axis s=iω via the open-loop transfer function — again, evaluating along e^{iθ} directions.

Bottom line: wherever something oscillates, rotates, or carries phase, Euler's formula replaces two real functions (sin, cos) with one complex exponential that adds angles when multiplied.

6. Interactive Lab works offline

Drag sliders. All rendering is done with the browser's Canvas API — no external libraries.

1) Unit circle explorer

θ = 1.00 rad (57.3°)

e^{iθ} = cosθ + i sinθ projection cosθ (real) projection sinθ (imag)

2) Taylor convergence

θ terms N

Partial sum S_N = Σ_{k=0}^{N} (iθ)^k/k!. Vectors head-to-tail spiral toward the true point on the circle.

3) Multiplication = rotation

θ₁ θ₂

z₁=e^{iθ₁}, z₂=e^{iθ₂}, z₁·z₂ = e^{i(θ₁+θ₂)}. Angles add.

4) Visual proof of e^{iπ} = −1

7. Octave / MATLAB examples

Copy-paste into Octave (open-source) or MATLAB. No toolboxes required.

7.1 Plot the unit circle via e^{iθ}

% plot_euler_circle.m
theta = linspace(0, 2*pi, 1000);
z = exp(1i*theta);          % Euler's formula
figure; plot(real(z), imag(z), 'LineWidth', 2); axis equal; grid on;
hold on; plot([-1.2 1.2], [0 0], 'k:', [0 0], [-1.2 1.2], 'k:');
title("e^{i\\theta} traces the unit circle");
xlabel('Re'); ylabel('Im');
% mark special points
pts = [0, pi/2, pi, 3*pi/2];
plot(cos(pts), sin(pts), 'ro', 'MarkerFaceColor','r');
text(cos(pts)+0.05, sin(pts)+0.05, {"0","\\pi/2","\\pi","3\\pi/2"});

7.2 Taylor series convergence

% taylor_convergence.m
theta = 2.0; Nmax = 20;
S = 0; z = 1i*theta;
term = 1;
figure; hold on; axis equal; grid on;
th = linspace(0,2*pi,400); plot(cos(th),sin(th),'k:');
for n = 0:Nmax
  if n>0, term = term*z/n; end
  S = S + term;
  plot(real(S), imag(S), 'bo-');
end
plot(cos(theta), sin(theta), 'r*', 'MarkerSize',10);
title(sprintf('Taylor partial sums to e^{i%.2f}, N=%d',theta,Nmax));
legend('unit circle','partials','true');

7.3 Roots of unity

% roots_of_unity.m
n = 7;
k = 0:n-1;
roots = exp(1i*2*pi*k/n);   % e^{i2πk/n}
figure; plot(real(roots), imag(roots), 'o', 'MarkerSize',8,'LineWidth',2);
hold on; plot([0 real(roots)],[0 imag(roots)],':'); axis equal; grid on;
title(sprintf('%dth roots of unity', n));

7.4 Fourier epicycle (square wave)

% fourier_epicycles.m
t = linspace(0, 2*pi, 2000);
x = zeros(size(t)); y = zeros(size(t));
N = 9; % odd harmonics
figure; hold on; axis equal; grid on; xlim([-2 2]); ylim([-2 2]);
for k = 1:2:2*N-1
  r = 4/(pi*k);
  x = x + r*cos(k*t);
  y = y + r*sin(k*t);
  % each term is r*e^{ik t} decomposed
end
plot(x, y, 'b', 'LineWidth',1.5);
title('Fourier sum of odd harmonics ≈ square wave via Euler');

These scripts embody the same idea: replace sin/cos pairs with a single complex exponential, then let geometry do the algebra.

Closing

Euler's formula is not a trick of notation. It is the definition of the complex exponential forced by demanding that the derivative remain itself, even when the input is imaginary. That demand turns growth into rotation, and the result is the circle.

From this single identity flow de Moivre's theorem, the roots of unity, Fourier series, phasors, quantum phases, and the mathematics of rotation. Learn to read e^{iθ} as "rotate by θ", and much of modern science becomes a one-liner.