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You've probably all seen the Euler theorem e^(pi*i)=-1 Me and one of my algebra II classmates have come up with some derivations of this equation for things like the logarithm of imaginary numbers; does anyone know how to take the log of a complex number, ie log (a+bi)?????? We're stuck here and I can't find a proof....
I do not know how to make Greek letters when posting to the forum, So I will use R and A instead of Rho and Theta, which are normally used for radial distance and angle in polar coordinates.
I will be a little sloppy and not use multiply operator with i (squareroot of minus one). Also, ln is usually used instead of log for natural logarithms.
The mathematics which leads to e^iPi = -1 is derived from complex numbers expressed in Polar coordinates.
R = SquareRoot(x^2 + y^2)
A = ArcTangent(y / x )
x + iy = R*e^iA, which is Polar form.
ln(x + iy) = ln(R*e^iA)
ln(x + iy) = ln(R) + ln(e^iA)
ln(x + iy) = ln(R) + iA
Sweet. Thanks man.