18 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-46.

Answer Problems 1–12 without referring back to the text. Fill in the blank or answer true/false.

  1. The sector defined by −π/6 < arg z < π/6 is a simply connected domain. _____
  2. If f(z) dz = 0 for every simple closed contour C, then f is analytic within and on C. _____
  3. The value of dz is the same for any path C in the right half-plane Re(z) > 0 between z = 1 + i and z = 10 + 8i. _____
  4. If g is entire, then where C is the circle |z| = 3 and C1 is the ellipse x2 + y2/9 = 1. _____
  5. If f is a polynomial and C is a simple closed curve, then f(z) dz = _____.
  6. If f(z) = where C is |z| = 3, then f(1 + i) = _____.
  7. If f(z) = z3 + ez and C is the contour z = 8eit, 0 ≤ t ≤ 2π, then dz = _____.
  8. If f is entire and |f(z)| ≤ 10 for all z, then f(z) = _____.
  9. dz = 0 for every simple closed contour C that encloses the points z0 and z1. _____
  10. If f is analytic within and on the simple closed contour C and z0 is a point within C, then
    dz._____

  11. where n is an integer and C is |z| = 1.
  12. If |f(z)| ≤ 2 on |z| = 3, then _____ .

In Problems 13–28, evaluate the given integral using the techniques considered in this chapter.

  1. (x + iy) dz;C is the contour shown in FIGURE 18.R.1
    A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a point on the negative horizontal axis labeled minus 4 and a point on the positive horizontal axis labeled 3. The graph shows a line, labeled C, starting at minus 4, moving vertically upward, then bending rightward at 90 degrees, then intersecting the vertical axis, then continuing straight, then turning downward at 90 degrees, and finally meeting the horizontal axis.

    FIGURE 18.R.1 Contour in Problems 13 and 14

  2. (x − iy) dz;C is the contour shown in Figure 18.R.1
  3. |z2| dz;C is z(t) = t + it2, 0 ≤ t ≤ 2
  4. eπz dz; C is the line segment from z = i to z = 1 + i
  5. eπz dz;C is the ellipse x2/100 + y2/64 = 1
  7. sin z dz;C is z(t) = t4 + i(1 + t3)2, −1 ≤ t < 1
  8. (4z3 + 3z2 + 2z + 1) dz; C is the line segment from 0 to 2i
  9. (z−2 + z−1 + z + z2) dz; C is the circle |z| = 1
  10. dz;C is the circle |z| = 2
  11. dz;C is the circle |z − 1| = 3
  12. dz;C is the circle |z| =
  13. is the ellipse x2/4 + y2 = 1
  14. z csc z dz;C is the rectangle with vertices 1 + i, 1 − i, 2 + i, 2 − i
  15. dz;C is the contour shown in FIGURE 18.R.2
    A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a point on the negative horizontal axis labeled minus 2 and a point on the extreme right of the positive horizontal axis labeled 3. A line segment, labeled C, starts at minus 2, moves vertically downward till the center of the third quadrant, then turns rightward to become horizontal, then intersects the negative vertical axis, then rises diagonally to pass through point 3 on the positive horizontal axis, then passes through point 3 and moves diagonally upward to intersect the positive vertical axis, and finally falls diagonally to merge with minus 2 on the negative horizontal axis.

    FIGURE 18.R.2 Contour in Problem 27

  16. dz; C is (a) |z| = 1, (b) |z − 3| = 2, (c) |z + 3| = 2
  17. Let f(z) = zng(z), where n is a positive integer, g(z) is entire, and g(z) ≠ 0 for all z. Let C be a circle with center at the origin. Evaluate dz.
  18. Let C be the straight line segment from i to 2 + i. Show that