Euler's Formula

Euler's formula is the following identity in complex analysis making use of the imaginary number $i$:

$e^{i\phi }=\cos \phi +i\sin \phi \!$

Euler's Identity

The formula can be used to derive an equation known as Euler's identity:

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

This is called the most beautiful equation in mathematics, because it equates the five most common mathematical constants in a perfectly elegant manner. Notice, however, its lack of rigor because it essentially depends on the above arbitrary definition for using the imaginary square root of -1 as an exponent, i.e., that $e^{i\phi }=\cos \phi +i\sin \phi \!$

See also

• De Moivre's Theorem
• Leonard Euler