### Homework page for Math 403:01, spring 2002

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) 5z2+4z+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 (z2) =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: x2 < y 8. H={z=x+iy: -Pi (less than or equal to)y < Pi}

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-z0|< r} is star-shaped.
(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<|2z2-12z+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. zn=((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 z0=8 and then at z0=8+i.

In Exercises 15 to 20, find all points of continuity of the given function.
15. f(z)= ((z3+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/(n2+in)

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

One definition of ab 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 ii, 22, i2, and 2i.

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.

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.