1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 337

u.,., - Uyy = 0. Show that W1 = Re u( z' z) and W2 = Im u( z, z) are

solutions of Wxy = 0 in the same neighborhood.

29. Let u( x, y) be real analytic in some neighborhood of the origin. Suppose
    that u satisfies in that neighborhood the partial differential equation
    Uxy = 0. Show that w1 = Reu(z,z) and w 2 = Imu(z,z) are harmonic
    functions in the same neighborhood.

30. More generally, if u(x, y) is real analytic in some neighborhood of the

origin, and a and b are real constants, prove the following:

(a) If u satisfies the elliptic differential equation

Uxx + Uyy + a(ux + uy) +bu= 0

then w1 = Re u( z, z) and w 2 = Im u( z, z) satisfy in the same

neighborhood the hyperbolic equation

w.,., - Wyy + 2aw., + 2bw = 0

(b) If u satisfies the hyperbolic equation

Uxy + a(u., + uy) +bu= 0

then w 1 = Re u(z, z) and w 2 = Im u( z, z) satisfy in the same

neighborhood the elliptic equation

w.,., + Wyy + 4aw., + 4bw = 0

( c) If u satisfies the hyperbolic equation

u.,., - Uyy·+ a(u., - uy) = 0

then w 1 = Reu(z,z) and w 2 = lmu(z,z) satisfy m the same

neighborhood the hyperbolic equation

Wxy +awy = 0 (L. Kiser [73])

31. If u = u(x,y) is harmonic and of class c<^3 >(D) in a domain D where

Uy # 0, prove that

w = Arctan u.,

Uy

is also harmonic in the same domain.

32. Investigate the existence of nonconstant harmonic functions having the
    form u = g(xy). Find the corresponding conjugate harmonic if such
    u does exist.


33. Suppose that the function u = u(x, y) is harmonic and homogeneous

of degree m # 0 in a region R. Show that

v = m-^1 (yu., -xuy)