# Analysis Incarnate

15 Aug 2020This is a quick proof of Euler’s formula without the use of the Taylor Series.

Even with the imaginary number `i`

, this equation is differentiable, so we can take the derivative of :

Using the quotient rule, we end up with :

The derivative of `f(x)`

is $ie^{i\phi}$, and the derivative of `g(x)`

is $e^{i\phi}$. Let’s focus on the numerator. Distributing the respective pairs, we get :

As you can see, the derivative of `f(x)=0`

, and the only function whose derivative is zero is a constant. Let’s call our constant `c`

. If we find a value for `c`

, we know the value is the same for the entire equation, because it’s a constant. If we plug “0” into the equation that we originally derived :

After simplifying, we end up with (1/1), because $e^{i0}$ is one, $isin(0)$ is zero, and $cos(0)$ is one. Since we know that `c`

is a constant, the value of `f(x)`

remains the same.

After we simplify, we are done :)