families of level curves
1. Sketch the families of level curves of u and v for the following func-
tions J = u + iv. (a) J(z) = liz, (b) J(z) = l/z2, (c) J(z) = z6.
Determine where J(z) is conformal and where it is not conformal.
2. Sketch the families of level curves of u and v for J(z) = Logz =
u + iv. Relate your sketch to a figure in Section 1.6.
3. Sketch the families of level curves of u and v for the functions J =
u+iv given by (a) J(z) = e Z , (b) J(z) = ea:z, where a is complex.
Determine where J(z) is conformal and where it is not conformal.
62 II Analytic Functions
4. Find a conformal map of the horizontal strip {-A < 1m z < A}
onto the right half-plane {Rew > O}. Hint. Recall the discussion of
the exponential function, or refer to the preceding problem.
5. Find a conformal map of the wedge {-B < arg z < B} onto the
right half-plane {Rew > O}. Assume 0 < B < 7r.
6. Determine where the function J (z) = z +1Iz is conformal and where
it is not conformal. Show that for each w, there are at most two
values z for which J(z) = w. Show that if r > 1, J(z) maps the circle
{Izl = r} onto an ellipse, and that J(z) maps the circle {Izl = I/r}
onto the same ellipse. Show that J(z) is one-to-one on the exterior
domain D = {Izl > I}.
Determine the image of D under J(z).
Sketch the images under J(z) of the circles {Izl = r} for r > 1, and
sketch also the images of the parts of the rays {arg z = .8} lying
in D.
7. For the function J(z) = z + liz = u + iv, sketch the families of
level curves of u and v. Determine the images under J(z) of the top
half of the unit disk, the bottom half of the unit disk, the part of
the upper half-plane outside the unit disk, and the part of the lower
half-plane outside the unit disk. Hint. Start by locating the images
of the curves where u = 0, where v = 0, and where v = 1. Note
that the level curves are symmetric with respect to the real and
imaginary axes, and they are invariant under the inversion z ~ liz
in the unit circle.
8. Consider J(z) = z + eiOl.lz, where 0 < 0: < 7r. Determine where
J(z) is conformal and where it is not conformal. Sketch the images
under J(z) of the unit circle {Izl = I} and the intervals (-00,-1]
and [+1, +00) on the real axis. Show that w = J(z) maps {Izl > I}
conformally onto the complement of a slit in the w-plane. Sketch
roughly the images of the segments of rays outside the unit circle
{argz = .8, Izl ~ I} under J(z). At what angles do they meet the
slit, and at what angles do they approach oo?
9. Let J = u+iv be a continuously differentiable complex-valued func-
tion on a domain D such that the Jacobian matrix of J does not
vanish at any point of D. Show that if J maps orthogonal curves to
orthogonal curves, then either J or 1is analytic, with nonvanishing
derivative.
