Homework for Math 421 Spring 2026

SHOW ALL YOUR WORK!!!

• Assignment 1   (Due Thursday, February 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 of graded problems
• Assignment 2   (Due Tuesday, February 17)
  • 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, February 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
  • 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).
• Assignment 4   (Due Thursday, March 5)
  • 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 Sec 31 #5b, Sec 32 #1     Sec 33 #7,9
• Assignment 5   (Due Thursday, March 12)
  • 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 Tuesday, March 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 ...)).
  • Solutions to Sec 38 problems 2(c), 3, 4: Solutions
  • Solutions to Sec 39 problem 2: Solutions
• Assignment 7   (Due Thursday, April 2)
  • Section 42 page 135:   1, 3, 5, 6, 7, 8, 10.
  • Solutions to Sec 42 problems 1,3,5,10: Solutions
• Assignment 8   (Due Thursday, April 9)
  • 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 9   (Due Thursday, April 16)
  • Section 49 page 160:   1 All except (d), 2 ALL, 3, 4, 6, 7.
  • Optional extra problem on Cauchy Goursat's Theorem for simply connected domains (need also Theorem 1 in Section 52)
  • Solutions to Sec 49 problems 2(b), 6: Solutions and : problem 7
• Assignment 10   (Due Tuesday, April 21, day of second midterm)
  • 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 11   (Due Thursday, April 30)
  • 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 12   (Due Thursday, May 7)
  • Section 56 page 188:   2, 3, 4
  • Section 59 page 195:   1, 2, 3, 6, 7, 8, 11(a).
  • 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 can be also found here:: Example
  • Section 71 page 239:   1 a,b,c,d (see hint below), 2 a,b,c, (residue at infinitey) 3 a, b, 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.
• Assignment 13   (Due Wednesday, May 13, day of the final, not to be handed in, but the material is covered in the final exam)
  • 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, 10.
  • Section 86 page 297:   (Argument Principle) 1, 2, 5.