Homework for Math 421 Fall 2016

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
  • A scanned copy of the problems for assignment 1:  
• 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.
• Assignment 3   (Due Thursday, September 29)
  • Section 21,22,23 page 71:   1 (c), (d), 2 (b), (d), 3 (a), 5, 6
  • 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
• Assignment 4   (Due Thursday, October 6)
  • Section 34 page 108:   2, 3, 5 (a), 10, 13, 16 (do problem 16 anyway you want, not necessarily using cosh and sinh).
  • 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.
• Assignment 5   (Due Thursday, October 13)
  • Section 32 page 100:   1, 2, 5, 6.
  • Section 33 page 104:   1, 2 (c), 3, 5, 6, 7, 9.
• Assignment 6   (Due Thursday, October 27)
  • 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: realize this function as the real part of an analytic function).
  • Section 38 page 121:   1 (b), 2 (b), (c), 3, 4, 5
  • Section 39 page 125:   1, 2
• Assignment 7   (Due Thursday, November 3)
  • Section 42 page 135:   1, 3, 5, 6, 7, 8, 10
• Assignment 8   (Due Thursday, November 10)
  • Section 43 page 140:   1, 2, 4, 7
  • Section 45 page 149:   1, 2 (a), (b), 3, 4, 5
  • Section 49 page 160:   1 All except (d)
• Assignment 9   (Due Tuesday, November 15 - DATE OF MIDTERM 2)
  • Section 49 page 160:   2 ALL, 3, 4, 6, 7.
• Assignment 10   (Due Thursday, November 17)
  • Section 52 page 170:   1 a, b, c, d, e, 2, 3, 4, 5, 7, 8, 10.
• Assignment 11   (Due Thursday, December 1)
  • Section 54 page 179:   1, 2, 3, 4, 5, 6, 7, 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).
• Assignment 12   (Due Thursday, December 8)
  • Section 56 page 188:   2, 3, 4
  • Section 59 page 195:   1, 2, 3, 6, 7, 8, 11, 13
• Assignment 13   (Due Tuesday, December 13 - Last class)
  • Section 62 page 205:   1, 2, 3, 4, 5, 6. Find also the Laurant series of the function f(z) 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.
  • Section 71 page 239:   1 ALL
    Hint for 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.
• Assignment 14   (Due December 22 - material will be covered on the final exam)
  • Section 71 page 239:   2 ALL, 3 ALL, 5, 6
    
  • 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), 5
  • Section 87 page 297:   Argument princimple: 1, 2, 5.
  • Section 87 page 297:   (Recommended, not covered in the final exam) Rouche's Theorem: 6, 7 (a), (c), 8, 9