Euler’s Formula and Exponential Form of Complex Numbers

Euler’s formula states that e^iθ = cos θ + i sin θ, which gives every complex number the exponential form z = re^iθ. In brief: e^iπ = −1, so e^iπ + 1 = 0 (Euler’s identity). In exponential form, multiplication is z₁z₂ = r₁r₂e^{i(θ₁ + θ₂)}, and powers are zⁿ = rⁿe^inθ. The number i = e^iπ/2 and −1 = e^iπ.

The exponential form compresses the polar form into one symbol and turns the rules for moduli and arguments into the ordinary laws of exponents. It connects the exponential function with trigonometry and is the standard notation for complex numbers in physics and engineering. This guide states the formula, shows why it is plausible via power series, derives the main consequences, converts between forms and gives a table of values of e^iθ.

What is Euler’s formula?

For every real number θ (in radians):

e^iθ = cos θ + i sin θ.

The number e^iθ lies on the unit circle at the angle θ. Its modulus is 1, since cos²θ + sin²θ = 1. Special cases: eⁱ⁰ = 1, e^iπ/2 = i, e^iπ = −1, e^i3π/2 = −i and e^i2π = 1. The special case θ = π gives Euler’s identity e^iπ + 1 = 0, which relates five fundamental constants: e, i, π, 1 and 0.

Why does it hold?

The power series of the exponential function, e^x = 1 + x + x²/2! + x³/3! + …, is also used to define e^ix for real x. Substitute x = iθ and use i² = −1, i³ = −i, i⁴ = 1:

e^iθ = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + iθ⁵/5! − …

Separate the real and imaginary terms: the real part is 1 − θ²/2! + θ⁴/4! − … = cos θ, and the imaginary part is θ − θ³/3! + θ⁵/5! − … = sin θ. These are the power series of cosine and sine. A second argument: the function f(θ) = e^iθ/(cos θ + i sin θ) has the derivative 0, so it is constant, and f(0) = 1.

What is the exponential form of a complex number?

Every nonzero complex number can be written as z = re^iθ, where r = |z| and θ is the argument. The conjugate is z̄ = re^−iθ. The laws of exponents apply:

• z₁z₂ = r₁r₂e^{i(θ₁ + θ₂)}
• z₁/z₂ = (r₁/r₂)e^{i(θ₁ − θ₂)}
• zⁿ = rⁿe^inθ
• e^iθ = e^{i(θ + 2π)}: the exponential is periodic with the period 2π.

What are the consequences?

• cos θ = (e^iθ + e^−iθ)/2 and sin θ = (e^iθ − e^−iθ)/(2i).
• The sum formulas follow from e^{i(a + b)} = e^iae^ib: comparing the real parts, cos(a + b) = cos a cos b − sin a sin b.
• The general complex exponential: e^{a + bi} = e^a(cos b + i sin b). Its modulus is e^a and its argument is b.
• The complex number iⁱ = (e^iπ/2)ⁱ = e^−π/2 ≈ 0.2079 is real.

Worked examples

Example 1: conversion to exponential form

1 + i = √2 e^iπ/4. And −2 = 2e^iπ. And −i = e^−iπ/2. And 1 − √3 i = 2e^−iπ/3.

Example 2: back to algebraic form

3e^i2π/3 = 3(cos 120° + i sin 120°) = −3/2 + (3√3/2)i ≈ −1.5 + 2.598i. And 2e^i0.5 = 2(0.8776 + 0.4794i) = 1.755 + 0.959i.

Example 3: a power

(1 + i)¹⁰ = (√2)¹⁰e^i10π/4 = 32e^i5π/2 = 32e^iπ/2 = 32i. Check with (1 + i)² = 2i: (2i)⁵ = 32 i⁵ = 32i ✓.

Example 4: an exponential with a complex exponent

e^{2 + iπ/3} = e²(cos 60° + i sin 60°) = 7.389(0.5 + 0.866i) = 3.695 + 6.399i.

Example 5: a trigonometric identity

cos²θ = ((e^iθ + e^−iθ)/2)² = (e^2iθ + 2 + e^−2iθ)/4 = (2 cos 2θ + 2)/4 = (1 + cos 2θ)/2, the power-reduction formula.

Example 6: solving an equation

Solve e^z = −1. Write z = x + iy: e^x(cos y + i sin y) = −1 = 1 · e^iπ. So e^x = 1, hence x = 0, and y = π + 2kπ. The solutions are z = i(2k + 1)π, infinitely many. This foreshadows the multi-valued complex logarithm.

Table of values of e^iθ

Where does the formula come from?

Leonhard Euler published the formula in his 1748 work Introductio in analysin infinitorum. Roger Cotes had found a related relation in logarithmic form in 1714. De Moivre’s formula of the early eighteenth century is the special case for integer powers. Euler’s identity e^iπ + 1 = 0 has often been called one of the most beautiful formulas in mathematics, in polls and in popular writing.

Where is it used today?

The Fourier transform expresses signals as sums of complex exponentials e^iωt, and the solutions of linear differential equations with oscillations are written with them. In quantum mechanics, the wave function of a free particle has the form e^{i(kx − ωt)}. In electrical engineering, a sinusoidal voltage V cos(ωt + φ) is the real part of V e^iφe^iωt, which turns calculus on sinusoids into algebra.

Which mistakes are common?

Using degrees in the exponent

In e^iθ, the angle θ must be in radians. The expression e^i90° is not i unless 90° is understood as π/2.

Forgetting the periodicity

e^iθ repeats every 2π. This is why the complex logarithm has many values.

Thinking that e^iθ is a real number

It is a complex number on the unit circle. Only for θ = kπ is it real (±1).

Mixing up the exponential form and the algebraic form

The sum e^iα + e^iβ has no simple exponential form. Convert to cos and sin first, or use the half-angle trick: e^iα + e^iβ = 2cos((α − β)/2) e^i(α+β)/2.