I Apparent counter-example to Cauchy-Goursat theorem (Complex Analysis)

[SomeBody]

TL;DR

It looks like the complex function 1/(z-p) does not comply with the Cauchy-Goursat theorem and the defination of the complex logarithm for certain values of z.

I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid.

I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem:

Let $f:D\to\mathbb{C}$ be an anlytic function over a simply connected region $D$. If $a$ and $z$ are part of $D$ then:
$$F(z)=\int_{z_{0}}^{z} f(w)dw$$
is analytic over $D$ and $F'(z)=f(z)$

Now the question:

If I pick $f(w)=\frac{1}{z-p}$ and a region $D$ with $z_{0}$ in it but without point $p$ enclosed (think some u-shape region stretched around p on one side) then $f$ is analytic over $D$ and I should be able to apply the theorem above. So:
$$F(z)=\int_{z_{0}}^{z} \frac{1}{z-p}dw=ln\left(\frac{z-p}{z_{0}-p} \right)$$

Now let $z_{1}=-z_{0}+2p$ be in $D$ which gives the following result:
$$F(z_{1})=ln\left(\frac{-z_{0}+2p-p}{z_{0}-p} \right)=ln\left(-1 \right)$$

However in the first chapter of my syllabus the complex logarithm is defined as:
$$Ln(z)=ln(r)+i\varphi$$ for every $z=re^{i\varphi}$ with $\varphi\neq \pi+2k\pi, k\in \mathbb{Z}$ and undefined for complex numbers with negative real part and zero imaginary part (like for the negative real numbers).

So if $F(z_{1})$ is undefined, how can $F$ be analytical over $D$ like the theorem states?

 

[fresh_42]

$\log(-1)=i\pi$
https://www.wolframalpha.com/input?i=log(-1)

 

[FactChecker]

Good question. If you want to keep the domain, $D$, simply connected, you can not include any point on the negative reals (the branch cut).

 

[SomeBody]

$\log(-1)=i\pi$
https://www.wolframalpha.com/input?i=log(-1)

Yes I think this is the (obvious) answer.
I was confused by the definition of $Ln(z)$ with branch cut on the negative real axis. In order to be analytic, a branch cut is necessary (and it does not have to be at$\pi$, its angle can be chosen) but in fact $Ln(z)$ is a multi-valued function which loops increasingly and does not stop at the negative real axis.
So I guess the solution to my question is $F$ is indeed the natural logarithm but with a branch cut at an angle different from $\pi$.

 

[mathwonk]

The natural logarithm is by definition any analytic function defined on a (connected) open set U not including 0, and (right-) inverse to the exponential function. (I.e. such that exp(Ln(z)) = z for all z in U.) Such inverses do not exist on open sets which include a loop that goes around the origin a non zero number of times. If U is any connected open set not including 0, and not containing any loop that encircles the origin, and if c is any point of U, then there are an infinite number of ways to define a natural logarithm in U. Namely, since the exponential function takes the value c infinitely many times, there are infinitely many choices of Ln(c). However, once a choice A of Ln(c) has been made, then Ln(z) = A + the integral of 1/z, from c to z, defines uniquely a branch of the natural logarithm everywhere in U.

In your notation, this is the case p = 0 and c = z0.

One simple way to choose a connected open set that does not contain a loop encircling the origin, is to let U be the complement of an infinite ray emanating from the origin. In that case too, one is still free to choose the value of Ln(z0) in infinitely many ways.

The complement of a ray is one example of a simply - connected set, but simple connectedness is vast overkill for defining a Log. In a simply connected set U, no loop in U can wind around any point outside the set U, whereas all that is needed to define Ln is that no loop in U should wind around 0. In general the (counter-clockwise) winding number around p of an oriented loop, is (1/2π) times the integral of dz/(z-p) around that loop. So if the integral of dz/z equals zero over every loop in a connected set U, then the integral of dz/z, integrated along any path from z0 to z in U, defines an analytic branch of Ln(z) - A, where A is any choice of Ln(z0), i.e. a branch of Ln(z) - Ln(z0) = Ln(z/z0).

The apparently very special example of the complement of a ray can be generalized to give essentially the most general simply-connected open set. I.e. a (not necessarily connected) open set U, not containing 0, in the complex plane, is apparently simply- connected if and only if its complement in the Riemann sphere is a closed, connected set, containing both 0 and infinity. In particular the closure in the Riemann sphere of a ray starting from the origin is such a closed connected set, as is the closure of any unbounded path starting from the origin. But not every closed connected set in the sphere, containing 0 and infinity, contains a path joining 0 to infinity, e.g. the closure of the topologist's sine curve does not.