Homework for Math 421 Fall 2022

SHOW ALL YOUR WORK!!!

• Assignment 1   (Due Thursday, September 15)
  • 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
  • Section 10 page 29:   2 (a), (b), 4, 7, 8
  • Solutions to the four graded problems: Solutions
• Assignment 2   (Due Thursday, September 22)
  • 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 29)
  • 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
  • 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).
  • Solutions to the four graded problems: Sec 25 #7, Sec 29 #4, 6     Sec 34 #16
• Assignment 4   (Due Thursday, October 6)
  • 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.
  • Section 26 page 81:   1 (a), (d), 2, 4
    
  • Solutions to the four graded problems: Sec 31 #5b, Sec 32 #1     Sec 33 #7,9, Sec 26 #1a
• Assignment 5   (Due Thursday, October 13)
  • Section 26 page 81:   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: realize this function as the real part of an analytic function).
  • Section 38 page 121:   1 (b), 2 (b), (c), 3, 4, 5 (postponed to next week, but should have been assigned this week)
  • 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 ...)).
  • Solution to Sec 31 problem 10: Solutions
  • Solutions to Sec 39 problem 2: Solutions
• Assignment 6   (Due Thursday, October 20)
  • Section 38 page 121:   1 (b), 2 (b), (c), 3, 4, 5
  • Section 42 page 135:   1, 3, 5, 6, 7, 8, 10
  • Solutions to Sec 38 problems 2(c), 3, 4: Solutions
  • Solutions to Sec 42 problems 1,3,5,10: Solutions
• Assignment 7   (Due Thursday, October 27)
  • Section 43 page 140:   1, 2, 4, 7
  • Section 45 page 149:   1, 2 (a), (b), 3, 4, 5
  • Solutions to Sec 43 problem 7, section 45 problems 3 and 5, section 49 problems 1 (e): Solutions
• Assignment 8   (Due Thursday, November 3)
  • Section 49 page 160:   1 All except (d), 2 ALL, 3, 4, 6, 7.
  • Solutions to Sec 49 problems 2(b), 6: Solutions and : problem 7
  • Extra problem on Cauchy Goursat's Theorem for simply connected domains (need also Theorem 1 in Section 52)
• Assignment 9   (Due Thursday, November 10)
  • Section 52 page 170:   1 a, b, c, d, e, 2, 3, 4, 5, 7, 10.
  • Solutions to Sec 52 problems 2 and 5: Solutions
  • 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
• Assignment 10   (Due Thursday, November 17)
  • Section 54 page 179:   3, 4, 5, 6, 7.
  • Solution to Sec 54 problems 3 and 6: Solution
  • Section 56 page 188:   2, 3, 4
  • Solution to Sec 56 problem 4: Solution
• Assignment 11   (Due Thursday, December 1)
  • Section 59 page 195:   1, 2, 3, 6, 7, 8, 11, 13.
  • Solution to Sec 59 problem 13: Solution
  • 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.
  • Solution to Sec 62 problem 2: Solution
  • Solution to the extra problem: Solution
• Assignment 12   (Due Thursday, December 8)
  • 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)
  • 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 13   (Not to be handed in, but material was covered in class and will be covered in the final. It will be assumed that you have solved these problems)
  • Section 76 page 255:   1, 2 (a), 3 (a), 4 (a), 5, 10.
  • Section 87 page 297:   (Argument Principle and Rouche's Theorem) 1, 2, 5, 6, 7 (a), (c), 8, 9.