Explanation on how we use polar coordinates/differential equations to prove Euler's formula, leading onto Euler's identity. Thanks
Will this do as a proof?
Wow, thank you so much. The math behind this elegant formula is amazing. Thanks once again!
ℕℙ
Let z be the complex number on the unit circle in the complex plane.
Thus, z= cos(φ)+isin(φ).
(1) iz = icos(φ)-sin(φ)
(2) dz/dφ = -sin(φ)+icos(φ)
Notice how (1) and (2) are equal to eachother.
(3) dz/dφ = iz
(3) is in the form f'(x) = a*f(x).
This equation implies that the derivative of f(x), f'(x), is equal to its original function f(x) when a=1
( f(x)=f'(x) ). Therefore, f(x)=e^x, when a=1.
Moreover, when f(x)*a =f'(x), the chain rule is applied. Thus, f(x)=e^(ax).
Going back to (3) one sees that a=i. Consequently, z=e^(iφ)
Finally, e^(iφ)=cos(φ)+isin(φ)
QED