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


We know the Maclaurin series of the functions



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


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




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


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


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)


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.


See also

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