CKP notation: 
; 
, independent variable;
z, dependent variable. I will use
; z, independent variable; w, dependent variable.
If f(z) is multi-valued, choose a single branch of the function and an appropriate z-domain to study and map.
Conformal mapping: Angle between intersecting curves in the z-domain is preserved under the mapping.
Figure 1: At points of analyticity, with 
mappings of complex
functions preserve angles.
Analytic function in the neighborhood of a point 
where
defines a conformal map in that neighborhood.
First consider differential arcs. Set
i.e., as 
If
That is, segments in the w-plane are just rotated by 
the argument of from their
pre-image inclination in the z-plane.
Furthermore, the segment 
is scaled in length by
under the papping.
Preservation of wedge angles follows from this result:
then,
angle of wedge is preserved. Now assume that Conclude that
Discuss.
m distinct directions in z map into same direction in w.
Inverse function, 
has m branches, one for each
choice of j. Inverse is a multi-valued function.
This discussion can be extended to poles and algebraic branch points, as well, in the following sense. Suppose that
Here, g(z) is assumed to be analytic in some punctured neighborhood of
and 
For 
, a = 0, otherwise, a and
b are real.
Then,
Hence, if z traverses an arc of small radius about
with opening angle,
, as in the figure, the image of that arc has opening angle
.