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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}} $

See also

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