I've been told by some students that the text is currently unavailable. So below are the problems students should hand in:

**Due Monday, January 28** Read sections 1.1. and 1.1.1!

The Entrance Exam

**Due Wednesday, January 30** Read section 1.2!

Section 1.1

2: Use the quadratic formula to solve these equations; express the
answers as complex numbers. (c) 5*z*^{2}+4*z*+1=0.

4. Find Re(1/*z*) and Im(1/*z*) if *z=x+iy*, with
*z* not equal 0. Show that
Re(*iz*)=-Im *z* and Im(*iz*)= Re *z*.

11. Show that |*z+w*|^{2}-|*z-w*|^{2}=4
Re(*z*{complex conjugate of *w*}) for any complex numbers
*z*,*w*.

Section 1.2

7. Describe the locus of points *z* satisfying the given
equation.
Re (*z*^{2}) =4.

21. Let *a* be a complex number with 0< |*a*|< 1. Show
that the set of all *z* with (b) |*z-a*|=|1-{complex
conjugate of *a*)*z*| is the circle
{*z*:|*z*|=1}.

(**Hint** Square both sides and
simplify.)

**Work requested in class**
Locate some complex numbers (see the
questions and the answers).

**Due Monday, February 4** Read section 1.3!

Section 1.2

Follow the technique outlined in the text to find all solutions to
the given equation.

24. *(z+1) ^{4}=1-i*.

Section 1.3

For each of the sets in Exercises 1 to 8, (a) describe the interior and
the boundary, (b) state whether the set is open or closed or neither
open nor closed, (c) state whether the interior of the set is
connected (if it has interior).

3. *C=*{*z=x+iy*: *x ^{2} < y*
8.

19. Determine which of the following sets are star-shaped.

(a) *D*={*z*: |*z-*1|<*2* or
|*z+*1|<*2*}.

(b) *D*={*z*: *x*>0 and |*z*|>1}.

**Note** To do problem 19 you will need the text of problem 18,
You need *not* hand in problem 18. Here it is:

18. An open set *D* is **star-shaped** if there is some point
*p* in *D* with the property that the line segment from
*p* to *z* lies in *D* for each *z* in
*D*.

(a) Show that the disc {*z*
:|*z-z _{0}*|<

(b) Show that any convex set is star-shaped.

**Work requested in class**
Suppose 1<|*z*|<2. Find positive numbers *A* and *B* so
that *A*<|2*z*^{2}-12*z*+1|<*B*. Finding
*B* is a straightforward use of the Triangle Inequality. For
*A* a use of the "reverse" Triangle Inequality is needed, with a
correct selection of the `BIG` and `LITTLE` terms. This
selection is somewhat delicate.

**On Monday, February 4**

**Work requested in class**
Draw some pictures (see the
questions and the answers).

**Due Monday, February 11** Read section
1.4!

Section 1.4

In Exercises 1 to 8, find the limit of each sequence that converges;
if that sequence diverges, explain why.

2. *z*_{n}=((1+*i*)/sqrt{2})^{n}

In Exercises 9 to 14, find the limit of each function at the given
point,or explain why it does not exist.

11. *f*(*z*)=(1- Im *z*)^{-1} at
*z*_{0}=8 and then at *z*_{0}=8+*i*.

In Exercises 15 to 20, find all points of continuity of the given
function.

15. *f*(*z*)=
((*z*^{3}+*i*)/(*z*-*i*)) if *z* is
not equal to *i* and *f*(*z*)=-3 if *z*=*i*.

19. *h*(*z*)=*z* if |*z*| less than or equal to 1
and *h*(*z*)=|*z*|^{2} if |*z*| > 1.

In Exercises 31 to 39, determine whether the given infinite series
converges or diverges.

36. The sum, as *n* goes from 1 to infinity, of
1/(*n*^{2}+*i*^{n})

**Due Wednesday, February 13** Read section
1.5!

Section 1.5: 11, 23, 28

**On Monday, February 11**

**Work requested in class**
Questions about exp & log (see the
questions and the answers).

**Due Monday, February 18** Finish section
1.5!

Section 1.5: 4, 8, 9, 19

Also please answer the following question:

One **definition** of *a*^{b} is exp(*b*
log *a*) (in this course we will use the complex analysis
meanings of exp and log!). Use this definition to find all possible
values of *i*^{i}, 2^{2},
*i*^{2}, and 2^{i}.

**Comment** The problem is "easy" but irritating. `Maple`
could help you with part of it. For example, the command
`evalf(I^I);` gets the response .2078795764 which might be an
approximation to part of one answer above. Note, though, that I'd like
exact answers in terms of traditional mathematical constants and
values of calc 1 functions, not approximations.

**Due Monday, February 25** Read section 1.6
carefully. The methods and ideas will be used constantly in the
course.

Section 1.6: 1, 2, 4, 5, 7, 15

**Due Wednesday, February 27** Read section 2.1
carefully. The methods and ideas will be used constantly in the
course.

Section 2.1: 1a, 6, 14, 16, 20c,e

Our **first exam** will be given in the
standard class time & place on **Wednesday March 6**. It will cover
up to whatever we get done during this Wednesday's class (February
27). I will attempt to have some review problems to give out, and I
will also attempt to schedule a review time before the first exam
(probably on the evening of Tuesday, March 7).

**For Monday, March 4** Read
section 2.2.

Please review for the exam. Get a copy of the review sheet and *do the problems!*. We will have a review
time, but there certainly will not be enough time to cover all of the
problems in detail.

I have reserved Hill 525 at 7:40 PM on Tuesday, March 5, for a
review session. The time was selected because almost everyone has a
class at almost all sensible times!

**Due Wednesday, March 13** Read section
2.3, which, together with 2.4, is the core of the subject.

Section 2.3: 1, 2, 4, 9, 10, 14

**Due Wednesday, March 27** Read
section 2.4.

Section 2.4: 1, 3, 5, 13, 17, 18, 20, 21

**No homework due on Wednesday, April 2!** You should be reading on
in chapter 2, please: isolated singularities, residues, and computations of
integrals.

**Due Monday, April 15** Read 2.6
and 3.1 These two sections will be covered on the second exam
which will
be given on Monday, April 22, at the standard class time and
place. Therefore you should get familiar with them.

Section 2.6: 3, 5, 9, 13, 21

**Due Wednesday, April 17** Read
3.1 The exam on Monday, April 22, will cover up to and including
section 3.1. If you wish to hand in solutions to the problems listed
below on Wednesday, I
will try to grade them Wednesday evening so that you can pick up
graded solutions on Thursday or Friday.

Section 3.1: 10, 15, 17a, c

Review problems for the second exam are available, as are hints and a complete exposition of the solutions.

**Due Wednesday, May 1** Read 3.3
and 3.4

Section 3.3: 4a,b, 5a,e, 7a

Review problems for the final exam are available. Review sessions will be held in Hill 525 on Wednesday, May 8, from 10 to 11:30 AM and on Thursday, May 9, from 3:30 to 5 PM.

**
Maintained by
greenfie@math.rutgers.edu and last modified 5/2/2002.
**