Analysis Incarnate

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

a

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

b Using the quotient rule, we end up with :

c

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 :

d

e

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 :

f

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.

g

After we simplify, we are done :)

h