Changes

*Thus, a logarithm of a complex number z is a complex number w such that <math>e^w = z</math>.

*Thus, a logarithm of a complex number z is a complex number w such that <math>e^w = z</math>.

*The notation for such a <math>w</math> is <math>ln z</math> or <math>log z</math>.

*The notation for such a <math>w</math> is <math>ln z</math> or <math>log z</math>.

*If <math>z = re^{iθ}</math> with <math>r > 0</math>which is in Polar form, then <math>w = ln r + iθ</math> is one logarithm of z.

*If <math>z = re^{i\theta}</math> with <math>r > 0</math>which is in Polar form, then <math>w = ln r + i\theta</math> is one logarithm of z.

*Adding integer multiples of 2πi gives all the others.

*Adding integer multiples of 2πi gives all the others.

*The complex exponential function is not injective, because <math>e^{w+2\pi i} = e^w</math> for any w, since adding iθ to w has the effect of rotating <math>e^w</math> counterclockwise θ radians.  

*The complex exponential function is not injective, because <math>e^{w+2\pi i} = e^w</math> for any w, since adding iθ to w has the effect of rotating <math>e^w</math> counterclockwise θ radians.  

*So the points <math>.....w-4\pi i,w-2 \pi i, w, w+2\pi i,w+4 \pi i....</math>

*So the points <math>.....w-4\pi i,w-2 \pi i, w, w+2\pi i,w+4 \pi i....</math>