Re: Question to Euler.

Op 26/05/2025 om 22:56 schreef Python:

Le 26/05/2025 à 22:30, Richard Hachel a écrit :

Z=r(cosθ+i.sinθ), I understand.

 Are you sure of that? I doubt it.
 

Z=e^iθ, I don't understand the expansion.

 No doubt about that. What you wrote is incorrect, the correct expression is Z = r*e^(iθ)
 Nice that you asked though.
 

Does anyone understand?

 Sure.
 sin(x) = sum_(k=0)^\infty ((-1)^k x^(1 + 2 k))/((1 + 2 k)!)
 cos(x) = sum_(k=0)^\infty ((-1)^k x^(2 k))/((2 k)!)
 e^x = sum_(k=0)^∞ x^k/(k!)
 i^2 = -1
 Hence:
 e^(ix) = cos(x) + i*sin(x)
 

https://youtu.be/FFPXm-tuOt8?t=263
We can kind of get a feeling for it by interactively observing the Taylor expansion (using the slider for N) to see how they approach the
transcendental functions with polynomials.
https://www.desmos.com/calculator/z2jjsuv5o2
https://youtu.be/3geVAJvJM8c?t=701