1550251515-Classical_Complex_Analysis__Gonzalez_

336 Chapter^6

where 8u/8s stands for the directional derivative of u in the

direction of the vector s, and similarly for the other derivatives.

21. Let w = f(z) = u +iv, z E D, D open.

(a) If lim Re(~ w / ~z) exists at a point of D, show that u,, and Vy

Az-+O

exist at that point and u., = Vy.

(b) If lim Im(~w/~z) exists at a point of D, show that Uy and v,,

Az-+O

exist at that point and Uy = -v,,.

( c) Suppose that u and v are differentiable at some point of D. Prove

that the existence of either of the limits in (a) and (b) implies the

existence of the other, and so the monogeneity off at that point.

22. Let f be analytic in a domain D. Prove that f is a constant in D if

either u =Ref, v = Imf, lfl or Argf is constant in D.

Also, prove that f is a constant in D if h(u, v) = aou^2 +a1uv+a2v^2 +

a 3 u + a 4 v is constant in D, the coefficients ai being complex constants.

23. (a) If the functions fk (k = 1,2, ... ,n) are analytic in a domain D,

and are such that

n

2: 11kczw = c (a constant)

k=I

at all points of D, prove that all functions fk reduce to constants

in D.

(b) Prove that w = 2:~ lfk(z)l^2 cannot be harmonic in D unless all the

analytic functions fk reduce to constants in D.

24. Suppose that f is analytic in a domain D and that f'(z) '=fa 0, lf(z)I < 1

at every point of D. Show that the function

lf'(z)I

w =Log 1 - lf(z)l2

satisfies in D the equation \7^2 w = 4e^2 w.

25. Let w = lf(z)l2, where f is analytic in a domain D. Show that w

. satisfies in D the equation w\7^2 w = w; + w~.

26. Let f be analytic in a domain D. Show that u = cos[Imf(z)] and

v = sin[Imf(z)] are solutions of the equation \7^2 1/J + Jf'(z)J^2 1/J = 0.

27. Let u(x, y) be a harmonic function analytic in the real sense about
    the origin [i.e., such that u( x, y) has a Taylor series representation in
    some disk with center at OJ. Show that both w 1 = Reu(z,z) and
    W2 = Im u(z, z) satisfy in a neighborhood of the origin the partial
    differential equation Wxx - Wyy = 0.


28. Let u(x, y) be real analytic in some neighborhood of the origin. Suppose
    that u satisfies in that neighborhood the partial differential equation