# Euler's Identity

**Type:** Theorem  
**Domain:** Named Results · Analysis  
**Codex URL:** /mathematics/named-results/eulers-identity/  
**Entry status:** Live — v1.0 (2026-05-27)

## Statement
e^(iπ) + 1 = 0

## Summary
Special case of Euler's formula eⁱˣ=cos(x)+i·sin(x) at x=π. Connects e, i, π, 1, 0. Often voted most beautiful equation in mathematics (Mathematical Intelligencer 1988, Physics World 2004).

## Euler's formula
eⁱˣ = cos(x) + i·sin(x), derivable via Taylor series substitution x→ix.

## Historical development
Roger Cotes (1714) found a related logarithmic identity but didn't develop exponential form. Euler derived it ~1740, published 1748 in *Introductio in analysin infinitorum*.

## Sources
### Tier 1
- Euler, L. (1748). *Introductio in analysin infinitorum*.
- Ahlfors, L.V. (1979). *Complex Analysis*. 3rd ed.
### Tier 2
- Maor, E. (1994). *e: The Story of a Number*. Princeton University Press.
### Tier 3
- Crease, R.P. (2004). The greatest equations ever. *Physics World*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*