Euler's Equation from Taylor series

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In this video, I showed how obtain to Euler's equation from the Taylor polynomials of exponential function, sine and cosine. This equation has been described as the most beautiful mathematical equation. I do not disagree.

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Prime Newtons is in a class by himself! Excellent job explaining this. ❤😊🎉

punditgi

I agree that this is the most beautiful equation in math. I saw it before, but it was done by Maclaurin series, which actually is a special case of Taylor series.
Good explanation as usual. 👍

EE-Spectrum

True, the Taylor series expansion was what led Euler to discover e^(ix) = cos(x) + i·sin(x). The way that you worked it out was exactly the same way that Euler himself worked it out, but did you know that you can prove e^(ix) = cos(x) + i·sin(x) without appealing to the Taylor series?
Any two functions that satisfy the same differential equation are equivalent to each other.
With that in mind, let f(x) = e^(ix) and let g(x) = cos(x) + i·sin(x).
Now, df/dx = i·e^(ix) = i·f(x), and dg/dx = - sin(x) + i·cos(x) = i·[cos(x) + i·sin(x)] = i·g(x). Since df/dx = i·f(x) and dg/dx = i·g(x), it follows that f(x) = g(x) ◼

johnnolen

been trying 2 think of an intuitive way to understand this without taylor series. I think it's because, if you have a point whose x and y coordinate's second time derivatives are minus themselves, it will create a circle (at least with some set of initial conditions). Just for geometrical reasons(?). So that gives you a pi. And differentiating something to get itself times a constant introduces e. Differentiating *twice* gives you that constant squared, and if that constant squared is minus 1, then the constant is i. So that's the link between e, i and pi

dantheman

TREMENDOUS VIDEO

NOTE:
💥TIME STAMP: 3:20 - 3:43:💥 eˣ = 1 + x + ½x² + ⅙x³ + ... + xⁿ/n! + ...

Greetings From Curaçao, An Island Nation in The Caribbean.

AngeloLaCruz

You have five important constants, addition, multiplication and exponentiation

holyshit

hey man im 16 years old and i realy like your videos

ali-htoe

What a nice vedio...
Understanding complex number makes it easy 😋

odumosuadeniyilukman

،،،،،،very very very well and thank you for

masoudhabibi

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