Prove Euler Identity without using Euler Formula

In summary: So, g is a curve in S1.If g were a curve of constant speed then1 = |g(L)| = |g(0)| = |g(L/2)|would imply that g(L/2) is a stationary point, so g(L/2)=1 or -1.If g(L/2) = 1 then g(t)=1 for all t, so g is a constant curve, so g=1 (since g(0)=1).If g(L/2)=-1 then g(t)=-1
  • #36
if you integrate the form dlog(z) = dz/z around the upper half of the unit circle, and get ipi, it seems to me you have proved that e^(ipi) = -1.

i.e. here e^z is defined as the inverse of the integral of dz/z.
 
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  • #37
How are you evaluating that integral?
 

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