Homework for Math 421 Spring 2015

SHOW ALL YOUR WORK!!!

• Assignment 1   (Due Thursday, January 29)
  • 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
  • A scanned copy of the problems for assignment 1:  
• Assignment 2   (Due Thursday, February 5)
  • Section 10 page 29:   2 (a), (b), 4, 7, 8
  • Section 11 page 31:   1 (a), (c), (e), 2, 3, 4 (a), (b), 5
  • Section 12 page 37:   2, 3, 4
  • Section 18 page 55:   5
  • Section 20 page 62:   1 (b), (d), 4, 8 (a), 9.
• Assignment 3   (Due Thursday, February 12)
  • 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, February 19)
  • 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
  • Section 32 page 100:   1, 2, 5, 6
  • Section 33 page 104:   1, 2 (c), 3, 5, 6, 7, 9
• Assignment 5   (Due Thursday, February 26)
  • Section 26 page 81:   1 (a), (d), 4, 7, 8
    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
• Assignment 6   (Due Thursday, March 5)
  • Section 39 page 125:   1, 2
  • Section 42 page 135:   1, 3, 5, 6, 7, 8, 10
• Assignment 7   (Due Thursday, March 12)
  • Section 43 page 140:   1, 2, 4, 7
  • Section 45 page 149:   1, 2 (a), (b), 3, 4, 5
• Assignment 8   (Due Thursday, March 26)
  • Section 49 page 160:   1 All except (d), 2 ALL, 3, 4, 6, 7.
• Assignment 9   (Due Thursday, April 2)
  • Section 52 page 170:   1 a, b, c, d, e, 2, 3, 4, 5, 7, 8, 10.
  • Section 54 page 179:   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).
• Assignment 10   (Due Thursday, April 16)
  • Section 54 page 179:   3, 4, 5, 6, 7.
  • Section 56 page 188:   2, 3, 4
• Assignment 11   (Due Tuesday, April 28)
  • Section 59 page 195:   1, 2, 3, 6, 7, 8, 11, 13
  • 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, 2 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 12   (Due on May 4, not to be handed in, but material is on the final)
  • Section 71 page 239:   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 79 page 267:   1, 2, 3, 4, 5. Recommended: 9.