1. Sketch the gradient vector fields Vu and Vv for (a) u+iv = eZ, (b)
u +iv = Logz.
2. Let a be a complex number, a 1= 0, and let fez) be an analytic
branch of za on C\( -00,0]. Show that f'(z) = af(z)/z. (Thus
f'(z) = aza-l, where we pick the branch of za-l that corresponds
to the original branch of za divided by z.)
3. Consider the branch of fez) = JZ(l- z) on C\[O, 1] that has posi-
tive imaginary part at z = 2. What is J'(z)? Be sure to specify the
branch of the expression for J'(z).
4. Recall that the principal branch of the inverse tangent function was
defined on the complex plane with two slits on the imaginary axis
by
‘T’ -IlL (1 +iZ).Lan z = -2′ og -1-‘- ,
z – zz z rt (-ioo,-i] U [i,ioo) .
Find the derivative of Tan-1 z. Find the derivative of tan- 1 z for
any analytic branch of the function defined on a domain D.
5. Recall that cos- 1 (z) = -ilog[z ± v’z2 -1]. Suppose g(z) is an
analytic branch of cos-1 (z), defined on a domain D. Find g'(z).
Do different branches of cos-1 (z) have the same derivative?
6. Suppose h(z) is an analytic branch of sin-1(z), defined on a do-
main D. Find h'(z). Do different branches of sin- 1 (z) have the
same derivative?
7. Let f (z) be a bounded analytic function, defined on a bounded
domain D in the complex plane, and suppose that fez) is one-to-54 II Analytic Functions one. Show that the area of f(D) is given by
Area(f(D)) = f!v1!'(zWdXd y .
8. Sketch the image of the circle {I z – 11 :::; I} under the map w = z2.
Compute the area of the image.
9. Compute
f!v1!'(zWdXdY,
for f(z) = Z2 and D the open unit disk {Izl < I}. Interpret your
answer in terms of areas.
10. For smooth functions 9 and h defined on a bounded domain U, we
define the Dirichlet form Du(g, h) by
Jr { [8 9 8h 8g 8h]
Du (g, h) = }u 8x 8x + 8y 8y dx dy.
Show that if z = f(() is a one-to-one analytic function from the
bounded domain V onto U, then
Du(g, h) = Dv(g 0 f, h 0 f).
Remark. This shows that the Dirichlet form is a “conformal invari-
ant.
