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1.4.12
2.2 q1418
Final exam
Homework 1
Homework 2
Images and preimages
Incomplete lecture 9 notes
List of most confusing topics
Midterm
Midterm Summer 2011
Mobius transformations
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1.
Find all complex numbers z satisfying
Do not use trigonometric or exponential functions in your answer.
Solution:
hence
2.
Is the set
open, closed or neither open nor closed? Is it connected? Sketch its boundary and state whether it is open, closed or neither open nor closed.
Solution:
The set is neither open nor closed. It is not connected. The boundary is the union of the circle
with the horizontal rays
and
The boundary is closed.
3.
Sketch the preimage of the set
under the map f(z)=z^2+1+i. Is it connected?
Solution:
The preimage is the union of the two sets
and
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Find all complex numbers z satisfying
Do not use trigonometric or exponential functions in your answer.
Solution:
hence
2.
Is the set
open, closed or neither open nor closed? Is it connected? Sketch its boundary and state whether it is open, closed or neither open nor closed.
Solution:
The set is neither open nor closed. It is not connected. The boundary is the union of the circle
with the horizontal rays
and
The boundary is closed.
3.
Sketch the preimage of the set
under the map f(z)=z^2+1+i. Is it connected?
Solution:
The preimage is the union of the two sets
and