Euler's Identity

Euler’s identity is often held up as the most beautiful single equation in mathematics:

$e^{i\pi} + 1 = 0$

In one line it binds five fundamental constants — $0$, $1$, $e$, $i$, $\pi$ — through the three most basic operations of arithmetic: addition, multiplication, and exponentiation. But beauty aside, the identity is a consequence of something deeper: the analytic continuation of the exponential function to the complex plane. Everything interesting about it flows from that single extension.

1. The underlying theorem: Euler’s formula

Euler’s identity is a special case of the more general Euler’s formula:

$\boxed{\, e^{ix} = \cos(x) + i\sin(x) \,}$

valid for every $x \in \mathbb{R}$ (indeed, every $x \in \mathbb{C}$). Setting $x = \pi$ yields $e^{i\pi} = -1$, which rearranges into the identity.

The starting point is the power series definition:

$e^z := \sum_{n=0}^{\infty} \frac{z^n}{n!}, \qquad z \in \mathbb{C}$

This series converges absolutely on all of $\mathbb{C}$ (by the ratio test), so $e^z$ is an entire holomorphic function. For real $z$ it agrees with the familiar exponential; for $z = ix$ we compute:

$$\begin{aligned} e^{ix} &= \sum_{n=0}^{\infty} \frac{(ix)^n}{n!} = \sum_{n=0}^{\infty} \frac{i^n x^n}{n!}\\ &= \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!} + i \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}\\ &= \cos(x) + i\sin(x) \end{aligned}$$

where in the second step we split the sum by parity of $n$ and used $i^{2k} = (-1)^k$, $i^{2k+1} = i(-1)^k$. The final identification uses the Taylor series for sine and cosine, which can themselves be taken as the defining series of those functions. No geometry is needed; the derivation is entirely algebraic.

2. The geometric picture

The point $e^{i\theta}$ traces the unit circle in the complex plane as $\theta$ varies over $\mathbb{R}$: at $\theta = 0$ we sit at $1$, at $\theta = \pi/2$ at $i$, at $\theta = \pi$ at $-1$ — Euler’s identity. The real and imaginary parts of this point are $\cos\theta$ and $\sin\theta$, which is why in many modern textbooks this unit-circle construction is used as the definition of the trigonometric functions. (See the interactive visualization below.)

3. Consequences in complex analysis

The exponential as a group homomorphism

The map $\varphi: (\mathbb{R}, +) \to (\mathbb{C}^*, \cdot)$ given by $\varphi(\theta) = e^{i\theta}$ is a group homomorphism (because $e^{i(\theta_1 + \theta_2)} = e^{i\theta_1}e^{i\theta_2}$, which itself follows from the exponential’s series identity). Its image is the unit circle $S^1 \subset \mathbb{C}^*$, and its kernel is $2\pi\mathbb{Z}$:

$\mathbb{R}/2\pi\mathbb{Z} \xrightarrow{\;\sim\;} S^1$

This is the Lie-group isomorphism between $\mathbb{R}$ (mod $2\pi$) and $U(1)$, the simplest non-trivial compact Lie group and the gauge group of electromagnetism. Euler’s identity simply says that $\pi$ maps to the antipode of $1$.

De Moivre’s theorem

Euler’s formula gives De Moivre’s theorem almost for free:

$(\cos\theta + i\sin\theta)^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)$

Using this, multiple-angle formulas and trigonometric identities become algebraic manipulations.

Roots of unity

The $n$-th roots of unity are exactly the solutions to $z^n = 1$ in $\mathbb{C}$, namely:

$\zeta_k = e^{2\pi i k / n}, \qquad k = 0, 1, \ldots, n-1$

These form the cyclic group $\mu_n \subset S^1$ of order $n$ and are foundational in algebraic number theory (cyclotomic fields), Fourier analysis (discrete Fourier transform), and cryptography.

4. Why physics can’t live without it

Harmonic oscillators

The solution to the undamped harmonic-oscillator equation $\ddot{x} + \omega^2 x = 0$ is most compactly written as $x(t) = A e^{i\omega t} + \bar{A} e^{-i\omega t}$, the real and imaginary parts giving the physical cosine and sine solutions. Electrical engineers use exactly this representation to analyze AC circuits — the “phasor” method.

Schrödinger’s equation

The time-dependent Schrödinger equation $i\hbar\partial_t \psi = H\psi$ has formal solution $\psi(t) = e^{-iHt/\hbar}\psi(0)$. The time-evolution of every quantum system is a complex exponential of the Hamiltonian, which is why unitary operators and the exponential map sit at the center of quantum mechanics.

Fourier analysis

The Fourier transform

$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x}\, dx$

decomposes a function into pure-frequency components $e^{2\pi i \xi x}$. Every one of those exponentials is a corollary of Euler’s formula. Without it, there is no clean theory of signals, filters, or differential equations.

5. The generalized formula on $\mathbb{C}$

Analytic continuation extends the identity to arbitrary complex arguments $z = x + iy$:

$e^z = e^{x+iy} = e^x(\cos y + i \sin y)$

This decomposition is the key to the complex logarithm and hence to essentially all of complex analysis. The multivaluedness of $\log$ — the fact that $\log(e^{i(\theta + 2\pi k)}) = \log(e^{i\theta})$ for every integer $k$ — is a direct expression of the $2\pi$ periodicity of Euler’s formula.

6. Historical note

Euler published the formula in 1748 in Introductio in analysin infinitorum, chapter 8. His derivation used the product representations of sine and cosine rather than the modern Taylor-series approach, but the content is the same. The specific identity $e^{i\pi} + 1 = 0$ was not isolated as a named “jewel” until much later — it appears in this packaged form most prominently in mid-20th-century writing (Feynman, Penrose, and others).

Further reading

• Rudin, Real and Complex Analysis, ch. 1 — power-series definition of the exponential on $\mathbb{C}$.
• Ahlfors, Complex Analysis, ch. 2 — analytic continuation and the logarithm.
• Stein & Shakarchi, Complex Analysis, ch. 1 — applied view via Fourier analysis.
• Lang, Complex Analysis, ch. 1 — the exponential as a group homomorphism and the exponential map of Lie groups.

θ (radians)

0.000

0 π/2 π 3π/2 2π

cos θ

1.0000

sin θ

0.0000

e^(iθ)

1.0000 + 0.0000 i

Set θ = π and you land at −1, which rearranges directly into Euler's identity: e^(iπ) + 1 = 0.

Frequently asked

Where does Euler's formula actually come from?

From comparing the Taylor series of exp, cos, and sin. Extending the real exponential function to complex arguments via its power series, the even-indexed terms assemble into cos(x) and the odd-indexed terms into i·sin(x). The identity e^(iπ)+1=0 is the special case x = π of the general formula e^(ix) = cos(x) + i·sin(x).

Is the extension from real to complex exponentials really well-defined?

Yes. The power series Σ zⁿ/n! converges absolutely on the entire complex plane, so it defines exp: ℂ → ℂ as an entire holomorphic function. Euler's formula is then a theorem about this analytic extension, not a new definition. The uniqueness of analytic continuation guarantees there is no other consistent extension.

What is the group-theoretic content of the identity?

The map θ ↦ e^(iθ) is a surjective group homomorphism from the additive group (ℝ, +) onto the multiplicative circle group S¹ ⊂ ℂ*, with kernel 2πℤ. The identity e^(iπ) = −1 reflects that π is the generator of the quotient ℝ/2πℤ's 'half-turn' element — geometrically, the antipode of 1 on the circle.

How does this connect to Fourier analysis?

Every function in L²(ℝ) can be decomposed into complex exponentials {e^(iωt)}. These exponentials are the eigenfunctions of the translation operator, and Euler's formula is precisely the bridge that shows why sines and cosines (the 'physical' oscillations) and complex exponentials (the 'mathematical' ones) encode the same information. The Fourier transform lives or dies on this identity.

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