Using Cauchy-Riemann, show...?
Let Ω ? ? be an open subset
Let f = u + iv: Ω --> ? be holomorphic
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.
Now think of f as a map from Ω ? ?2 to ?2
Let J_f denote the Jacobi matrix of f
| f ' (z) | 2 = | det J_f (x,y) |
for z = x + iy ∈ Ω.
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),
By the Cauchy-Riemann Equations we can rewrite this as
Hence, det J_f = (u_x)2 + (u_y)2 = |f '(z)|2.
I hope this helps!