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  • Re: Question to Euler.

    From Python@21:1/5 to Nice that you on Mon May 26 20:56:17 2025
    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)

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  • From Richard Hachel@21:1/5 to All on Mon May 26 20:30:50 2025
    Z=r(cosθ+i.sinθ), I understand.
    Z=e^iθ, I don't understand the expansion.
    Does anyone understand?

    R.H.

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  • From sobriquet@21:1/5 to All on Tue May 27 02:31:36 2025
    Op 26/05/2025 om 22:30 schreef Richard Hachel:
    Z=r(cosθ+i.sinθ), I understand.
    Z=e^iθ, I don't understand the expansion.
    Does anyone understand?

    R.H.

    Some people claim that you're not supposed to understand math.
    All you can hope for is to get used to it.

    I think John Von Neumann said something along those lines.

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  • From sobriquet@21:1/5 to All on Tue May 27 04:54:26 2025
    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

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