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^(iphi)
, and the derivative of g(x)
is e^(iphi)
. 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 :)