assignments
due Thu Feb 4
assignment 1
Let z
be a complex number unless stated otherwise.
- Prove that | Re(z) | ≤ | z |
and | Im(z) | ≤ | z |
.
- Solve | z − 4 | = | z + 1 |
.
- Sketch and describe the set of complex numbers satisfying | z / 2 − 1 | = 2
. Please do not use a calculator.
- Sketch and describe the set of complex numbers satisfying Re(2 / (z − 1)) = 1
. Again, please do not use a calculator.
- Convert the following complex numbers into polar form:-
-
;
-
.
- Prove that | z1z2 | = | z1 | | z2 |
. Hint: show that | z1z2 | 2 = | z1 | 2 | z2 | 2
.
- Suppose that z1
and z2
are non-zero complex numbers. Prove that if | z1 + z2 | = | z1 | + | z2 |
, then z1 / z2
is a positive real number.
due Thu Feb 18
assignment 2
Do the following questions from the course text, Fundamentals of Complex Analysis with applications to Engineering & Science by E. B. Saff & A. D. Snider.
- Exercises 1.5 (on p37): 7, 19.
- Exercises 1.6 (on page 42): 2 - 8, 11.
- Exercises 1.7 (on page 50): 1, 6, 7.
Challenge questions: not necessarily part of the assignment but attempting these questions will improve your grade.
- Ex 1.6: 13.
- Ex 1.7: 2, 3, 4.
due Thu Mar 11
assignment 3
- Exercises 2.2 (on p63): 3, 6, 11, 21.
- Ex 2.3 (on p70): 2, 7, 9, 14, 15.
- Ex 2.4 (on p 77): 1, 2, 5.
- Use x = rcos(θ)
and y = rsin(θ)
together with the chain rule, for example
| ∂u | | ∂u | ∂x | | ∂u | ∂y |
| = |
|
| + |
|
|
| ∂r | | ∂x | ∂r | | ∂y | ∂r |
is one instance for
,to prove the Cauchy-Riemann equations for polar coordinates using the Cauchy-Riemann equations for cartesian coordinates.
Mid-Term Exams
due Thu Apr 15
assignment 4
- Ex 4.2 (on p170): 3, 5, 10, 12.
- Ex 4.3 (on p178): 1, parts (a) to (e).
due Thu Apr 29
assignment 5
- Ex 5.1 (on p239): 2, 7, 8, 9, 11, 12, 14.
due Tue May 4
End of term exam
due Tue May 11 9:00 am
Final exam
