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Let $\mathbb{R}^\leq$ be the set of real numbers that are less than or equal to zero.

I am thinking about the map $f(z) : \mathbb{C}\setminus\mathbb{R}^\leq \rightarrow \mathbb{C}$ given by $f(z) = \log(z)$. I know that the logarithm is analytic on this domain, therefore it must be conformal, which means the mapping has to preserve angles.

This implies some geometric properties of the space after the mapping is applied to it. For example, any triangle must have angles that add up to $180^\circ$, or $\pi$. I think it also means that any theorems of Euclidean geometry that are true of regular points in $\mathbb{C}\setminus\mathbb{R}^\leq$ will also be true after applying the log transformation, but I'm not sure about this, nor sure how to prove it if it is true.

I am able to do computations, for example, if I wanted to find all possible values of $\log(1+i)$, then I would do the following:

  1. Find the polar representation of $1+i$. The angle is $45^\circ$ which is $\pi/4$. The magnitude is $\sqrt{2}$. Therefore I can write $$1+i = re^{i\theta} = \sqrt{2}e^{i\pi/4}$$

  2. The angle doesn't have to be only $\pi/4$, we can also add or subtract any multiple of $2\pi$ and still get the same result: $$1+i = \text{any element of this set: } \left\{\sqrt{2}\exp\left(\displaystyle\frac{i\pi}{4} + 2k\pi i\right)\,\, {\Big | } \,\, k \in \mathbb{Z}\right\}$$

  3. This corresponds to the logarithm having infinitely many values: $$\log(1+i) = \text{any element of this set: }\,\, \boxed{\,\left\{\left(k+\frac{1}{8}\right)i \pi \ln 2 \,\,\,{\Big | } \, \,k \in \mathbb{Z}\right\}\,}$$

This method will also allow the computation of the infinitely many possible values for exponents like $\left(1+i\right)^{\left(1+i\right)}$.

I can do the computations and I can apply the definitions. However, I fundamentally lack any visual intuition about this mapping. The "Riemann surface" pictured on Wikipedia makes no sense to me and the explanation they give is very technical.

Any help giving me visual intuition for this mapping would be very useful. What does the space even look like after the log mapping is applied? Which theorems in Euclidean geometry of $\mathbb{R}^2$ are also true in $\mathbb{C}$ after the log mapping is applied? I know that the internal angles of any triangle add up to $180^\circ$ but I'm not sure about other theorems or if all of them hold.

Vincent Luo
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    This might help: https://math.stackexchange.com/questions/136570/determination-of-complex-logarithm?rq=1 – Jeanba Dec 01 '19 at 18:03
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    @ZubinMukerjee Not so sure, how would f(-1)=log(-1) be defined? Log is not defined for negative real values. – Jeanba Dec 01 '19 at 18:07
  • Yes, I was wrong - I will revert the edit. – Zubin Mukerjee Dec 01 '19 at 18:08
  • @Jeanba It depends on the branch cut. – Paul Frost Dec 02 '19 at 09:30
  • We have $w = \ln z = \ln |z| + i \arg z$. To make $\ln z$ analytic on $D = \mathbb C \setminus (-\infty, 0]$, we have to take $\arg z \in (-\pi + 2 \pi k, \pi + 2 \pi k)$ for $z \in D$. Then $\operatorname {Re} w \in (-\infty, \infty), , \operatorname {Im} w \in (-\pi + 2 \pi k, \pi + 2 \pi k)$, and the mapping from $D$ to this horizontal strip is bijective. – Maxim Dec 07 '19 at 00:47

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