Explanation on how we use polar coordinates/differential equations to prove Euler's formula, leading onto Euler's identity. Thanks

Discussion

Today I was trying to find the value of 2^i, not exact value though but something in the form of a formula. After 3 hours I ended up with a generalized formula for n^i. I am attaching an image and a pdf file of my work.

Will this do as a proof?

Wow, thank you so much. The math behind this elegant formula is amazing. Thanks once again!

@Aman NAGU

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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