Homework for Math 421 Fall 2024
SHOW ALL YOUR WORK!!!
- Assignment 1 (Due Thursday, September 12)
- Section 2 page 5: 4
- Section 3 page 8: 1 (a), (b)
- Section 4 page 12: 4, 5 (a), (c), 6
- Section 5 page 14: 1 (c), (d), 9, and the
extra problem
- Section 8 page 22: 1 (a), 2, 3, 4, 5 (c), 5
(a), 6, 9, 10. Hint for 9:
- Section 10 page 29: 2 (a), (b), 4, 7, 8
- Solutions to the four graded problems: Solutions
- Assignment 2 (Due Thursday, September 19)
- Section 11 page 31: Read this section and
answer the following questions: 1 (a), (c), (e), 2, 3, 4 (a), (b), 5
- Section 12 page 37: 2, 3, 4
- Section 18 page 55: 3, 5
- Section 20 page 62: 1 (b), (d), 4, 8 (a), 9.
- Section 21,22 page 71: 1 (c), (d), 2 (b),
(d), 3 (a), 5, 6.
- Solutions to the four graded problems: Sec 12
#4, Sec 18 #5, Sec 20 #9, and Sec
23 #2(d)
- Assignment 3 (Due Thursday, September 26)
- Section 25 page 77: 1 (a), (b), 2 (a), 3,
4(b), 7 (use the Theorem on page 74 and the cauchy Riemann equations).
- Section 29 page 92: 1, 3, 4, 6, 7, 8 (b), (c), 10, 11, 14
- Solutions to the four graded problems: Sec
25 #7, Sec 29 #4, 6
- Assignment 4 (Due Thursday, October 3)
- Section 34 page 108: 2, 3, 5 (a), 10, 13, 16
(do problem 16 anyway you want, not necessarily using cosh and sinh,
the answer in the text uses also the fact that (2+sqrt{3})(2-sqrt{3})=1).
- Section 31 page 97: 1, 2, 3, 5 (b), 7, 9. For
9 use the textbook definition of Log(z) on page 94, which is defined at
all points in the complex plane other than at 0. Recall that
with the textbook definition Log(z) is discontinuous at every negative real number.
- Section 32 page 100: 1, 2, 5, 6.
- Section 33 page 104: 1, 2 (c), 3, 5, 6, 7, 9.
- Solutions to the graded problems: Sec
31 #5b, Sec 32 #1 Sec 33
#7,9
- Assignment 5 (Due Thursday, October 10)
- Section 26 page 81: 1 (a), (d), 2, 4, 7, 8, 9.
Suggestion for 7: Either use the suggestion in the text, or
recall from calculus III that the tangent line to the curve
u(x,y)=constant, at a given point (a,b) on that curve, is
parallel to the line u_x(a,b)x+u_y(a,b)y=0, provided at least
one of the partials u_x(a,b) or u_y(a,b) does not vanish.
- Section 31 page 97: 10 (Method 1: check that
its first and second partials are continuous and
satisfies the laplace equation. Method 2: For every non-zero complex number c realize this function as the real part of an analytic function in some open set containing c).
- Solutions to: Sec 26 #1a
- Solution to Sec 31 problem 10: Solutions
- Assignment 6 (Due Thursday, October 24)
- Section 38 page 121: 1 (b), 2 (b), (c), 3, 4, 5
- Section 39 page 125: 1, 2, 6 (in part a, if
z(x) is on the x-axis, then y(x)=0, so z(x)=x, so show that
this is the case when x=1/n (n=1, 2 ...)).
- Section 42 page 135: 1, 3, 5, 6, 7, 8, 10.
- Solutions to Sec 38 problems 2(c), 3, 4: Solutions
- Solutions to Sec 39 problem 2: Solutions
- Solutions to Sec 42 problems 1,3,5,10: Solutions
- Assignment 7 (Due Thursday, October 31)
- Section 43 page 140: 1, 2, 4, 7
- Section 45 page 149: 1, 2 (a), (b), 3, 4, 5
- Solutions to Sec 43 problem 4 (the number of the
solution says 7, but it should be 4), section 45 problems 3 and 5: Solutions
- Assignment 8 (Due Thursday, November 7)
- Assignment 9 (Due Thursday, November 14)
- Section 52 page 170: 1 a, b, c, d, e, 2, 3, 4, 5, 7.
- Solutions to Sec 52 problems 2 and 5: Solutions
- Assignment 10 (Due Thursday, November 21)
- Section 52 page 170: 10.
- Solution to Sec 52 problem 10: Solution
- Section 54 page 179: (these three problems depend on section 53 only) 1, 2, 9.
- Additional problem: Let f be an entire function. Suppose that |f(z)|>2,
for all complex numbers z. Show that f must be a constant
function. Hint: consider g(z)=1/f(z).
- Solution to the additional problem: Solution
- Section 54 page 179: 3, 4, 5, 6, 7.
- Solution to Sec 54 problems 3 and 6: Solution
- Assignment 11 (Due Tuesday, November 26)
- Section 56 page 188: 2, 3, 4
- Solution to Sec 56 problem 4: Solution
- Section 59 page 195: 1, 2, 3, 6, 7, 8, 11, 13.
- Solution to Sec 59 problem 13: Solution
- Assignment 12 (Due Tuesday, December 3)
- Section 62 page 205: 1, 2, 3, 4, 5, 6. Extra problem: Find
also the Laurant series of the function f(z)=z/(z-1)(z-3) in problem 6 in each
of the three annular domains centered at the origin in which it is
analytic (unit disk, the annulus 1<|z|<3, and the complement
of the disk of radius 3). Hint: Use partial fractions to write f(z)
as A/(z-1) + B/(z-3) and treat each summand seperately. A
similar example was worked out in class and can be also found here:: Example
- Solution to Sec 62 problem 2: Solution
- Solution to the extra problem: Solution
- Assignment 13 (Due Tuesday, December 10)
- Section 71 page 239: 1 ALL (see hint below),
2 ALL, (residue at infinitey) 3 ALL, 5, 6.
- Hint for Section 71 problem 1d: Note that cot(z)sin(z)=cos(z).
Write cot(z)=sum_{n=-infinity}^infinity c_n z^n.
We already know the Taylor series of sin and cos centered at 0.
First show that c_n=0 for n<-1. Then solve for the coefficients c_{-1}, c_0, c_1, c_2, c_3 one by one each time expressing
c_{n+1} in terms of the previous ones c_{-1}, c_0, …, c_n.
- Section 72 page 243: 1 ALL, 2, 3, 4
- Section 74 page 248: 1, 2 (a), (b), 3, 4, 6 (a)
- Section 76 page 255: 1, 2 (a), 3 (a), 4 (a).
- Solution to Sec 71 problem 1(a),(c): Solution
- Solution to Sec 71 problem 2(a),(c), Sec 72 problem 1,
Sec 74 problem 4: Solution
- Assignment 14 (material is included in the final
exam, but it will not be collected. Give priority to the starred problems)
- Section 76 page 255: 5, 10.
- Section 87 page 297: (Argument Principle
and Rouche's Theorem) *1, *2, 5, *6, 7 (a), *(c), *8, 9.
- Section 79 page 267: 1, *2, *3, *4.