Euler's formula

Euler's formula is an important formula that links exponentials to trigonometric functions. It is given as follows,

e^{i \theta}=\cos \theta+i \sin \theta

To derive the formula, start from the exponential series with $z=i \theta$ ,

e^{i \theta}=1+i \theta+\frac{(i \theta)^{2}}{2 !}+\frac{(i \theta)^{3}}{3 !}+\cdots

Note that odd powers are imaginary while even powers are real. Separating odd and even powers,

\begin{aligned} e^{i \theta} & =\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 \end{aligned}

the real part is the Maclaurin series for $\cos \theta$ and the imaginary part is the Maclaurin series for $\sin \theta$ .

Using Euler's formula, the polar form of a complex number $z=r(\cos \theta+i \sin \theta)$ can be shortened to

z=r e^{i \theta}

An amusing example is $z=-1$ , for which $r=1, \theta=\pi$ , so that $-1=e^{i \pi}$ or

1+e^{i \pi}=0

This is sometimes referred to as Euler's equation which remarkably connects the five most famous numbers $0,1, e, \pi$ , and $i$ . Recall the following limit:

\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}

We can use the above equation to obtain Euler's equation by defining $e^{i \pi}$ as the limit of $(1+i \pi / n)^{n}$ . As $n$ gets large this quantity approaches -1 . The mathematical constant $e$ is also known as Euler's number.

Euler's formula can be used to derive the trigonometric addition formulae. Starting from

e^{i(\alpha+\beta)}=e^{i \alpha} e^{i \beta}

and applying Euler's formula to each exponential,

\begin{aligned} \cos (\alpha+\beta)+i \sin (\alpha+\beta) & =(\cos \alpha+i \sin \alpha)(\cos \beta+i \sin \beta) \\ & =\underbrace{(\cos \alpha \cos \beta-\sin \alpha \sin \beta)}_{\cos (\alpha+\beta)}+i \underbrace{(\cos \alpha \sin \beta+\sin \alpha \cos \beta)}_{\sin (\alpha+\beta)}, \end{aligned}

where the real part is the cosine addition formula while the imaginary part is the sine addition formula.

De Moivre's theorem

De Moivre's theorem, which is given as follows

(\cos \theta+i \sin \theta)^{n}=\cos (n \theta)+i \sin (n \theta)

with $n \in \mathbb{Z}$ , follows from Euler's formula,

\left(e^{i \theta}\right)^{n}=e^{i n \theta}

For $n=2$ , this gives the double angle formulae for the sine and cosine.

Euler's formula expresses an exponential in terms of trigonometric functions; one can also do the reverse, i.e. express $\cos \theta$ and $\sin \theta$ in terms of exponentials. By making use of

e^{i \theta}=\cos \theta+i \sin \theta

We can get

e^{i \theta}+e^{-i \theta}=2 \cos \theta,

which gives

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

Similarly, we obtain,

\sin \theta=\frac{e^{i \theta}-e^{-i \theta}}{2 i}

Euler's equation is often cited as an example of deep mathematical beauty. Stanford University mathematics professor Keith Devlin has said, "like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence". And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty".

The mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics". And Benjamin Peirce, a noted American 19th century philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth".

Definitions Complex functions