## Thursday, July 14, 2011

### Euler's Gem 2

Here is a sketch of a second derivation of Euler's famous formula:

 `eiθ = cosθ + i sinθ`

as presented by William Dunham in his book Euler, The Master of Us All. First post here.

The first step is to recall a standard trigonometric substitution in calculus:

 `y = sin xx = sin-1 y√(1 - y2) = cos x`

We're interested in the integral:

 `∫ dy / √(1 - y2)`

Substituting with x we see that:

 `dy = cos x dx`

And the integral is

 `∫ (1/cos x) cos x dx = ∫ dx = xx = ∫ dy / √(1 - y2)`

Now Euler makes a complex change of variable:

 `y = izx = ∫ dy / √(1 - y2) = ∫ i dz / √(1-(iz)2) = i ∫ dz / √(1 + z2) = i ln [√(1 + z2) + z]`

The last step is another standard result from calculus which I will assume for the time being (more here).

Undo the substitution:

 `z = y/i = sin x / iz2 = -sin2 x√(1 + z2) = √(1 - sin2 x) = cos xx = i ln (cos x + sin x / i)`

We will use two identities involving i:

 `u / i = - i u1 / (cos u - i sin u) = (cos u + i sin u)`

(For the second one, see the previous post). Now:

 `x = i ln (cos x + sin x / i)x = i ln (cos x - i sin x)ix = - ln (cos x - i sin x) = ln [ 1 / (cos x - i sin x) ] = ln (cos x + i sin x)`

Just eponentiate:

 `eix = cos x + i sin x`

Wow, again!