Suppose that fz ux y ivx y is analytic in some domain D in

Suppose that f(z) = u(x, y) + iv(x, y) is analytic in some domain D in the complex plane and that, for some point Z_0 = x_0 + iy_0 in D, we have f\'(z_0) notequalto 0. Let c_1 = u(x_0, y_0) and c_2 = v(x_0, y_0) a. Show that the families of level curves u(x, y) = c_1 and v(x, y) = c_2 are orthogonal. b. Give an example to show that the condition f\'(z_0) notequalto 0 is needed.

Solution

a]Since f(z) is analytic function, it satisfy C-R equations:

u_x = v_y and u_y = - v_x

=i/x+j/y
u=i u/x+j u/y=i v/y-j v/x

v=i v/x+j v/y
u.v=v/y v/x-v/x v/y=0
Therefore u = c₁ and v = c₂ cut orthogonally.

b] example: Linear transformation W =f(z) =aZ + b

f\'(z) = a = 0 if a=0

it will not be a conformal mapping.