Question

Using Cauchy-Riemann, show...?

Let Ω ? ? be an open subset

Let f = u + iv: Ω --> ? be holomorphic

1)

Show that ?z ∈ Ω, ?v(z) can be obtained by rotating ?u(z) by π/2.

In particular, ?u(z) and ?v(z) have the same modulus and are orthogonal to each other.

2)

Now think of f as a map from Ω ? ?2 to ?2

Let J_f denote the Jacobi matrix of f

Show that

| f ' (z) | 2 = | det J_f (x,y) |

for z = x + iy ∈ Ω.

Answer

1) ?u = <u_x, u_y>, and

?v = <v_x, v_y> = <-u_y, u_x> by Cauchy-Riemann Equations.

Therefore, ?u · ?v = <u_x, u_y> · <-u_y, u_x> = 0.

Hence, ?u and ?v are orthogonal

Moreover, ||?u|| = √((u_x)2 + (u_y)2) = ?v.

-----------------

2) Writing f(z) = u(x, y) + iv(x, y),

J_f =

[u_x u_y]

[v_x v_y].

By the Cauchy-Riemann Equations we can rewrite this as

[u_x u_y]

[-u_y u_x].

Hence, det J_f = (u_x)2 + (u_y)2 = |f '(z)|2.

I hope this helps!

Oct 10 at 15:15