Euler's formula

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

• De Moivre's formula