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What is the ln (0)??

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SsEe
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Zhen Lin
After reading some articles about the complex logarithm, it seems that there is indeed no way to have a meaningful value for ln 0. If we consider all the solutions to ew=ze^w = z and plot (z, w), it forms a pretty spiral sheet around z = 0, where it is discontinuous... Interestingly, if we think of the structure of it that way, it's hardly a surprise that we can arrive at different values for ln -1 by following different "paths" starting from z = 1.


The reason you get this is because the argument of a complex number is only defined up to multiplies of 2pi.

A complex number with argument θ\theta can be written z=∣z∣eθi+2kπiz=|z|e^{\theta i + 2k\pi i} where k is any integer.

Taking logs gives log⁡z=log⁡∣z∣+(θ+2kπ)i\log{z} = \log{|z|} + (\theta + 2k\pi) i. (where log⁡∣z∣\log{|z|} is real). So taking the log of a number gives lots of different answers, all separated by integer multiples of 2*pi*i.

To get a single valued function you just restrict the argument of z to some interval of length 2pi. This is a "branch" of log.

z=0 is a singularity (called a branch point).
Well, technically, if e^ix = 0 , then isin(x)=-cos(x), so itan(x)=-1

this implies that tan(x)=i, so x=arctan(i)

Putting this back in Euler's equation

e^(i*arctan(i)) = 0...

So ln(0)=i*arctan(i)

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