# Analysis Incarnate 2

This is a proof of Euler’s Identity using the Taylor Series.

$\dpi{300}&space;\tiny&space;e^{i\pi}+1=0$

We can write $$e^{i\phi}$$ as the Taylor Series expansion.

$\dpi{300}&space;\tiny&space;e^{i\phi}=\sum_{n=0}^{\infty}\frac{(i\phi)^{n}}{n!}$

Let’s visualize this sum. It helps to write out all your terms because you know that the powers of i are going to alternate between 1,-1,i,and -i.

# Analysis Incarnate

This is a quick proof of Euler’s formula without the use of the Taylor Series.

$\dpi{300}&space;\tiny&space;e^{i\phi}&space;=&space;cos&space;\phi&space;+&space;i&space;sin\phi$

Even with the imaginary number i, this equation is differentiable, so we can take the derivative of :

$\dpi{300}&space;\tiny&space;f(x)=\frac{e^{i\phi}}{cos\phi+isin\phi}$