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One of the most important identities in all of mathematics, Euler's formula relates complex numbers, the trigonometric functions, and exponentiation with Euler's number as a base.

The formula is simple, if not straightforward:

$\cos(\theta)+i\sin(\theta)=e^{i \theta}$
Alternatively:$\text{cis}(\theta)=e^{i \theta}$

When Euler's formula is evaluated at

$\theta=\pi$ , it yields the simpler, but equally astonishing Euler's identity.

As a consequence of Euler's formula, the sine and cosine functions can be represented as

$\sin(\theta)=\frac{e^{i \theta}-e^{-i \theta}}{2i}$
$\cos(\theta)=\frac{e^{i \theta}+e^{-i \theta}}{2}$

Derivation

We know the Maclaurin series of the functions

$e^x=\sum_{k=0}^\infty\frac{x^k}{k!}=1+x+\frac{e^2}{2!}+\frac{x^3}{3!}+\cdots$
$\sin(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$
$\cos(x)=\sum_{k=0}^\infty(-1)^k\frac{x^{2k}}{(2k)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$

Therefore

$e^{i \theta}=1+i\theta+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\frac{(i\theta)^4}{4!}+\frac{(i\theta)^5}{5!}+\cdots$
$=1+i\theta-\frac{\theta^2}{2!}-\frac{i\theta^3}{3!}+\frac{\theta^4}{4!}+\frac{i\theta^5}{5!}-\cdots$
$=\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\cdots\right)+i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\cdots\right)$
$=\cos(\theta)+i\sin(\theta)$

Trigonometry

The formula permits the extension of the trigonometric functions to complex-valued domains and ranges. In other words, it is possible to find complex, unrestricted $x$ values for which

$\cos(x)=a+bi$

and

$\cos(a+bi)=x$

In addition, it permits the determination of complex-valued inputs for values outside of the normal range of the trigonometric functions. In other words, if $x$ is complex, then it is possible that

$\cos(x)=2$

even though the domain of the cosine function is normally restricted to the real interval $[-1,1]$.

Exponentials

Exponential functions having a complex value in the exponent can also be evaluated:

$e^{a+bi}=e^a\cdot e^{bi}=e^a\cdot\text{cis}(b)=e^a\cdot\big(\cos(b)+i\sin(b)\big)$

Applications

Euler's formula is used extensively in complex analysis. It is also used often in differential equations, as Euler's number being raised a complex variable appears fairly often.

An interesting corollary of Euler's formula is that $i^i$ can be found and is entirely real.

$i^i=\left(\text{cis}\left(\tfrac{\pi}{2}\right)\right)^i=\left(e^{\tfrac{\pi}{2}i}\right)^i=e^{-\tfrac{\pi}{2}}$

However, $i^i$ does not have one singular representation, as the $\text{cis}(\theta)$ function is multivalued and depends on which branch is chosen. The general form for any integer $n$ is

$i^i=\left(\text{cis}\left(\tfrac{\pi}{2} + 2\pi n\right)\right)^i=\left(e^{\tfrac{\pi}{2}i +2\pi ni} \right)^i=e^{-2\pi n-\tfrac{\pi}{2}}$