Euler's Formula: the most beautiful math ever!

Euler's Formula

$\left.e^{i\theta }=\cos(\theta )+i\sin(\theta )\right.$

is the most beautiful math ever! Here are some of the reasons:

$e^{i\pi }+1=0$

$\left.e^{i\pi }=\cos(\pi )+i\sin(\pi )\right.$ so

$\left.e^{i\pi }+1=0\right.$

That alone is sufficient: a simple, elegant, lovely little equation linking five of the most important mathematial constants in the universe:

1. 1 -- no doubt the #1 number
2. 0 -- add 0 and you can write any number, using binary. So all the rest are just icing on the cake!
3. $\pi$ -- Babylonians, Archimedes -- a rich past!
4. $i$ -- imaginary, but not unimportant.
5. $e$ -- Euler proved irrational in 1737; Hermite proved transcendental in 1873.

Links exponentials to sines and cosines

Exponentials -- one of math's most important modeling tools for growth and decay of natural systems -- are linked to sines and cosines -- math's workhorses for periodic and oscillatory phenomena. The Taylor Series for sines and cosines fall right out of the Taylor series for the exponential, which is simply too marvelous for words (one needs some symbols and equations...)!

$\left.e^{i\theta }=\sum _{k=0}^{\infty }{\frac {(i\theta )^{k}}{k!}}\right.$

By identifying real and imaginary parts of this, we get the expansions for sine and cosine:

$\left.\cos(\theta )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\theta ^{2k}}{(2k)!}}\right.$

$\left.\sin(\theta )=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\theta ^{2k+1}}{(2k+1)!}}\right.$

Is a treasure trove of trig identities

for those of us who eschew memorization (for lack of a memory, say!). This permits us to memorize only a few things and to deduce the rest. The key is Euler's formula, via

$\left.e^{i(\theta _{1}+\theta _{2})}=e^{i\theta _{1}}e^{i\theta _{2}}\right.$

re-expressed using Euler's formula:

$\left.\cos(\theta _{1}+\theta _{2})+i\sin(\theta _{1}+\theta _{2})=(\cos(\theta _{1})+i\sin(\theta _{1}))(\cos(\theta _{2})+i\sin(\theta _{2}))\right.$

Multiplying out, we have a pair of identities

$\left.\left[\cos(\theta _{1}+\theta _{2})\right]+i\left[\sin(\theta _{1}+\theta _{2})\right]=\left[\cos(\theta _{1})\cos(\theta _{2})-\sin(\theta _{1})\sin(\theta _{2})\right]+i\left[\left(\cos(\theta _{1})\sin(\theta _{2})+\sin(\theta _{1})\cos(\theta _{2})\right)\right]\right.$

(once we identify real and imaginary parts).

$\left.\cos(\theta _{1}+\theta _{2})=\cos(\theta _{1})\cos(\theta _{2})-\sin(\theta _{1})\sin(\theta _{2})\right.$

$\left.\sin(\theta _{1}+\theta _{2})=\cos(\theta _{1})\sin(\theta _{2})+\sin(\theta _{1})\cos(\theta _{2})\right.$

From these all the usual trig identities can be derived (half angle, double angle, etc.).

De Moivre's theorem

$\left(\cos x+i\sin x\right)^{n}=\cos \left(nx\right)+i\sin \left(nx\right)$

can also be simply and elegently derived, via

$\left.e^{i(nx)}=\cos(nx)+i\sin(nx)\right.$