What does a parallel slit domain look like? After talking about the Riemann Mapping Theorem we also saw that doubly connected subdomains of the complex plane are also conformal to some standard domain - an annulus. I asked whether multiply connected domains are also conformal to some standard domain. He casually answered that.. 

.. every $n$-connected subdomain of the plane is conformal to some
  parallel slit domain where the boundary consists of $n$ parallel lines with specifiable slope.

Im not sure if these were exactly his words. Apparently I couldn't find anything in my books or online. Thus I tried to draw some domains of this shape on my own. But I failed. First I drew the whole plane with $n$ cuts. This is still simply connected. When the cuts are infinitely long the domain is no longer connected. 
How can the boundary consists of $n$ parallel lines such that the domain is still $n$-connected? Can you draw a picture for me or give me a hint? 
 A: This an explanation without proofs (see for example Nehari, Conformal Mapping for a reasonably simple account)
The case of $2$-connected domains is simple since there is essentially only one parameter - wlog one can assume the two complement connected components are non-degenerate (not points) as otherwise, the domain is trivially conformal to either the punctured plane or the punctured unit disc by filling in and using the usual Riemann mapping theorem, and then the domain is conformally equivalent to an annulus that can be taken to have the unit disc as the outer boundary with the radius of the inner circle as the parameter 
However when $n \ge 3$ there are $3n-6$ real parameters, so while the result above is still true in the sense that one can map the domain onto a domain bounded by analytic Jordan curves (or points if some components of the complement are degenerate) with the outer boundary being the unit circle (proof by induction using RMT and filling in components of the complement first all but one so we get a simply connected one and then removing them one by one carefully), there is no easy way to decide when such domains are conformally equivalent one to another as in the annulus case, so it was found that other domains work better as there are canonical maps to domains as mentioned in the OP with the most common being parallel slit, circular slit, radial slit, circle with concentric circular slits or circular ring with concentric circular slits as in all these cases standard unique maps can be given with canonical unicity data 
The parallel $n$ slit domain is the extended plane (so including $\infty$) from which we cut $n$ parallel (Euclidean) segments; the unicity data is then a point $w \in D$ that is mapped to infinity by a conformal map with expansion near $w$ given by $f(z)=\frac{1}{z-w}+a_1(z-w)+a_2(z-w)^2...$ and the angle the $n$-slit segments make with the real axis, while the parameters are the length of the slits and their initial points (note that initial points have $2$ real coordinates, so those and length give $3n$ parameters as the fixed angle determines them uniquely then, and  we have fixed pole residue taken $1$ - locally univalent requires simple pole by geometry - fixed constant-coefficient taken as $0$ and obviously the point $w$ itself, so $6$ real degrees of freedom giving the $3n-6$ real parameters in the end)
Similar considerations apply for the other types of canonical domains.
