Keep angles and bend regions

Mathematics · Complex Analysis · ages 21-22

Name ______________________ Date ____________

The tripling map sends one plus i to three plus three i. What is the real part of the image?
Answer: ______________
Why is the exponential map conformal at every point?
- It is a polynomial
- Its derivative e to the z never vanishes
- It maps lines to lines
- It is a bounded function
Why is the exponential map conformal at every point?
- It is a polynomial
- Its derivative e to the z never vanishes
- It maps every line to a line
A Mobius map is fixed once you know where three distinct points go.
Circle one: True False
The map w equal (z minus one) divided by (z plus one) sends one to zero. What is w at z equal to three?
Answer: ______________
Which Mobius map sends the upper half plane onto the unit disc with i going to 0?
- z plus i over z minus i
- z over z plus one
- z minus i over z plus i
Where does angle preservation fail for the squaring map w equal z squared?
- At 0, where the derivative vanishes
- Nowhere: squaring is conformal everywhere
- At infinity only
- At every point of the plane
If the derivative at a point is nonzero, the map preserves angles between curves through that point. True or false?
Circle one: True False
A Mobius map carries the real line somewhere. What can its image be?
- Circles and lines go to circles and lines
- All shapes become straight lines
- Angles double at every point
How do conformal maps help solve boundary value problems?
- They straighten every boundary into a line
- They remove all singularities automatically
- They make every solution constant on the boundary
- Map an awkward domain to a nice one, solve there, pull the solution back

Answer key

For grown-ups. Fold this page away before handing over the rest.

Keep angles and bend regions W1-mt_WtHCbAT4GI-s1

3 · Three times one plus i is three plus three i, whose real part is three.
Its derivative e to the z never vanishes · Nonzero derivative everywhere means angle preservation everywhere.
Its derivative e to the z never vanishes · A nowhere zero derivative means angle keeping at every point.
True · Three complex freedoms need three point images to fix.
0.5 · (Three minus one) divided by (three plus one) is two divided by four, which is 0.5.
z minus i over z plus i · It kills i, rounds the real line onto the unit circle, and keeps upstairs inside.
At 0, where the derivative vanishes · The derivative 2z is zero at the origin, where angles double instead of surviving.
True · True. Locally the map is rotation plus scaling, both angle preserving.
Circles and lines go to circles and lines · That circle to circle habit is what makes boundary transfer possible.
Map an awkward domain to a nice one, solve there, pull the solution back · Conformality preserves harmonicity, so solutions transfer between domains.