a) Since

as

, it makes sense to consider

(i.e. the integral converges). In addition,

.

b) Symmetry of the image: Let

. Considering the radial path

from 0 to

, we get

. Thus, it suffices to prove that one quarter of disc is sent to one quarter of square.

c) Of course,

. For

, we even have

, hence

.

d) Let's consider the image of the arc of circle from 1 to

. For

, we have (considering the path from 0 to 1 and then to

along the unit circle)

. And

hence

. Thus

simplify and

. Then

and the integral is real. Thus

for some

. This shows that the image of the arc of circle is a line segment (with slope

) from

to

.

I showed that the boundary of

is sent to the boundary of the square with vertices

, and 0 is sent to itself. Then there must be a connectivity argument to conclude...

In fact, this problem shows an example of Schwarz-Christoffel mapping ; there is a general formula alike for mapping the unit disc to any polygon.