20 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-51.

Answer Problems 1–10 without referring back to the text. Fill in the blank or answer true/false.

  1. Under the complex mapping f(z) = z2, the curve xy = 2 is mapped onto the line _____.
  2. The complex mapping f(z) = −iz is a rotation through _____ degrees.
  3. The image of the upper half-plane y ≥ 0 under the complex mapping f(z) = z2/3 is _____.
  4. The analytic function f(z) = cosh z is conformal except at z = _____.
  5. If w = f(z) is an analytic function that maps a domain D onto the upper half-plane v > 0, then the function u = Arg(f(z)) is harmonic in D. _____
  6. Is the image of the circle z − 1 = 1 under the complex mapping T(z) = (z − 1)/(z − 2) a circle or a line? _____
  7. The linear fractional transformation T(z) = maps the triple z1, z2, and z3 to _____.
  8. If f′(z) = z−1/2(z + 1)−1/2(z − 1)−1/2, then f(z) maps the upper half-plane y > 0 onto the interior of a rectangle. _____
  9. If F(x, y) = P(x, y) i + Q(x, y) j is a vector field in a domain D with div F = 0 and curl F = 0, then the complex function g(z) = P(x, y) + iQ(x, y) is analytic in D. _____
  10. If G(z) = ϕ(x, y) + iψ(x, y) is analytic in a region R and V(x, y) = , then the streamlines of the corresponding flow are described by ϕ(x, y) = c. _____
  11. Find the image of the first quadrant under the complex mapping w = Ln z = logez + i Arg z. What are images of the rays θ = θ0 that lie in the first quadrant?

In Problems 12 and 13, use the conformal mappings in Appendix D to find a conformal mapping from the given region R in the z-plane onto the target region R′ in the w-plane, and find the image of the given boundary curve.

  1. Two graphs. In the first graph, the horizontal axis is labeled x and the vertical axis is labeled y. The graph shows two points labeled A at the top of vertical axis and B at the bottom of the vertical axis. The first quadrant is shaded and labeled R. In the second graph, the horizontal axis is labeled u and the vertical axis is labeled v. The graph shows a point labeled i at the center of the positive vertical axis. A horizontal line intersecting at i in the second quadrant. The first and second quadrants are shaded and labeled R prime.

    FIGURE 20.R.1 Regions R and R′ for Problem 12

  2. Two graphs. In the first graph, the horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a point labeled 1 on the horizontal axis. A horizontal line passes through the horizontal axis from the origin to 1, a vertical line passes through the vertical axis from the origin to the top, and a vertical line intersecting the horizontal axis at 1 labeled A at the top and B at the bottom. The region bound by the lines is shaded and labeled R. In the second graph, the horizontal axis is labeled u and the vertical axis is labeled v. The graph shows a circle centered at the origin passing through the positive horizontal axis at 1. The region bound by the circle is shaded and labeled R prime inside it.

    FIGURE 20.R.2 Regions R and R′ for Problem 13

In Problems 14 and 15, use an appropriate conformal mapping to solve the given Dirichlet problem.

  1. A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows a point labeled 1 on the horizontal axis. A horizontal line passes through the horizontal axis from the origin to the extreme right and an increasing line from the origin to the top right forming an acute angle. A point labeled e^i pi over 4 is shown on the increasing line. u = 1 is labeled at the left of 1 and u = 0 is labeled at the right of 1 and u = 1 is labeled at the bottom of e^i pi over 4 and u = 0 is labeled at the top of e^i pi over 4. The region bound by the angle is shaded and labeled R.

    FIGURE 20.R.3 Dirichlet problem in Problem 14

  2. A graph. The horizontal axis is labeled x and the vertical axis is labeled y. The graph shows two circles internally tangent at the origin. The smaller circle intersecting the vertical axis at i and the bigger circle intersecting the vertical axis at 2 i. The boundaries of the bigger circle lie in the first and second quadrants are labeled u = 1 and the boundaries of the smaller circle lie in the first and second quadrants are labeled u = 0. The interior of the big circle and exterior of the small circle is shaded and labeled R.

    FIGURE 20.R.4 Dirichlet problem in Problem 15

  3. Derive conformal mapping C-4 in Appendix D by constructing the linear fractional transformation that maps 1, −1, ∞ to i, −i, −1.
    1. Approximate the region R′ in M-9 in Appendix D by the polygonal region shown in FIGURE 20.R.5. Require that f(−1) = u1, f(0) = πi/2, and f(1) = u1 + πi.
      A graph. The horizontal axis is labeled u and the vertical axis is labeled v. The graph shows a point labeled pi i over 2 at the center of the vertical axis. A horizontal line intersecting the vertical axis from the second quadrant to the first quadrant and a horizontal line intersecting the vertical axis at pi i over 2 in the first quadrant. A line with negative slope from the horizontal axis at u subscript 1 intersecting the vertical axis to the center left of the second quadrant at point labeled minus u subscript 1 plus pi i over 2 and an increasing line from the same point in the second quadrant intersecting the vertical axis to the horizontal axis in the first quadrant at point labeled u subscript 1 plus pi i over 2. An increasing line from pi i over 2 to u subscript plus pi i over 2 and a line with negative slope from u subscript 1 to pi i over 2 forming a left arrow. The regions are shaded differently and labeled R prime outside the arrow.

      FIGURE 20.R.5 Image of upper half-plane in Problem 17

    2. Show that when u1 → ∞,
    3. If we require that Im(f(t)) = 0 for t < −1, Im(f(t)) = π for t > 1, and f(0) = πi/2, conclude that
    1. Find the solution u(x, y) of the Dirichlet problem in the upper half-plane y ≥ 0 that satisfies the boundary condition u(x, 0) = sin x. [Hint: See Problem 6 in Exercises 20.5.]
    2. Find the solution u(x, y) of the Dirichlet problem in the unit disk z ≤ 1 that satisfies the boundary condition u(eiθ) = sin θ.
  6. Explain why the streamlines in Figure 20.6.5 may also be interpreted as the equipotential lines of the potential ϕ that satisfies ϕ(x, 0) = 0 for −∞ < x < ∞ and ϕ(x, π) = 1 for x < 0.
  7. Verify that the boundary of the region R defined by y2 ≥ 4(1 − x) is a streamline for the fluid flow with complex potential G(z) = i(z1/2 − 1). Sketch the streamlines of the flow.