Math 300, Fall 2023, Homework 4, Prof. Sachs Due, Friday, Nov. 10
Complete the following problems (or do as much as you can and show your
best attempt). Typed work is best, but clear handwriting is ok.
1. Write out a careful proof that the mapping z 7→ 1/z (a) will send circles
in the domain that pass through the origin into lines in the range that do not
pass through the origin in the range; and (b) the same mapping will also send
circles that do not pass through the origin in the domain onto circles in the
range that do not pass through the origin in the range.
2. Write down the degree 1 rational function that sends the points 1, i, −1 to
0, 1, ∞ respectively. Then consider the following questions:
(a) What is the image of the unit circle under this mapping?
(b) If you parameterize the unit circle in the usual way, what happens in the
range using this parameterization?
(c) What is the image of the parameterized line x = 1, y = t?
3. Consider degree 1 polynomial mappings of the general form: w = az + b
with a 6= 0. Suppose you are given two points in the domain and their
corresponding images, call them (z1, w1) and (z2, w2). How would you use
that to determine a and b?
4. Recall the definition of subgroup we have used earlier. Find all degree 1
rational functions that send ∞ to ∞. Show that these form a subgroup.
5. From your answer to the previous problem, find all mappings that not only
map ∞ to itself but also map 0 to 0. Show that these are a subgroup of the
whole set of mappings also.
6. (Challenge) Generalize the previous results to a very abstract statement: If
we have a group of 1:1, onto mappings from some set S onto itself, then
for any subset of elements within S, the subset of mappings that leave each
element of the subset unchanged, will form a subgroup of the full group of
mappings.