The neighborhood of infinity is viewed as a single point
under the mapping, 
.
Discuss the point at infinity for the mappings, 
and
, by discussing the neighborhood of the origin in the
-plane.
The mapping, 
.
Require, 
so that w(0) = i. (Why?)
Note that for large |z|,
, so that the z-plane, subject to these argument restrictions
maps into the w-plane with some local distortion near the origin
in z. The distortion diminshes with distance from the origin.
In Figure 8, we depict the image of two lines, just above(below)
the real axis in the z-plane. The turns through in these image lines
occur near w = 0, which is the image of
the branch points at 
The arguments of
of on the lines in the z-plane
change by nearly 
there and, hence, the arguments of
change by nearly 
In each case, only the argument of one
the square roots changes raplidly, while the other argument
remains nearly constant.
At z = 0, 
w'(0) = 0, and 
Hence, when 
changes by nearly
, 
changes by nearly 
(Note that I have not been specific about whether the argument of z changes by
postive or negative amounts; that depends on the direction of traversal of the
contours in the z-plane and also on whether one is speaking of the contour
passing above or below the point of interest.
It is good practice to carry out these traversals one's self and understand the image contours in more detail than is presented here. The image here is produced by Mathematica. In practice, I do not use lines above or below the axes, but lines on the axes. Then, near the branch points or zeroes of the derivative, I introduce a semi-circular arc around the point of interest; I know that a right angle turn away from the point produces a right angle turn in the image. Then, one only has to take account of the multiple change in angle along the arc according to whether the arc in the z-plane passes around a zero, pole branch point or essential singularity.
Note also that when a direction is assigned to a contour in the z-plane, points
to the right or left of that contour are mapped to the right or left of the image
contour, with a direction assigned to that image consistent with the direction of
the original contour. Thus, in this example, if one traverses the line above the
z-axis from 
to 
the image is the upper contour in Figure
8 traversed in the same direction. Hence, points above the contour
in the z-plane, the upper half z-plane, map to the upper half w-plane, above
the upper contour of the figure.
