Proving Analytic Function Convergence Using Constant Sum of Squares

[FanofAFan]

Homework Statement

Let f₁(z),... f_n(z) n >1 be analytic functions on a domain D such that |f₁ (z)|2 +...+|f_n(z)|2 = k (k is constant). Verify that all the functions must reduce to constants

Homework Equations

The Attempt at a Solution

Having the hardest time proving this, not sure where to being...

 

[snipez90]

What tools do you have to work with? I can probably think of a few ways to do this. Do you know what a harmonic function is?

 

[FanofAFan]

You mean harmonic function in terms of being analytic, or something else?

Also i was given the hit that |f_k(z)|² (|f₁(z)|²Delta f₁(z)|²)

 

[snipez90]

Well a harmonic function can be defined independently of analytic function theory, but yes the real and complex parts of an analytic function are harmonic, i.e., they satisfy http://en.wikipedia.org/wiki/Laplace's_equation#Laplace_equation_in_two_dimensions".

The original equation can be written in terms of the real and complex parts of each f_i (i = 1, ..., k). Differentiating the original equation twice with respect to x gives a new equation, call it (1). Differentiating the original equation twice with respect to y (use symmetry here), we get another equation (2). What happens when you add equations (1) and (2)?

 

[FanofAFan]

So would (1) be \partial_x (|f₁(z)|² +...|f_k(z)|²?

 

[snipez90]

Well no, you should have two x's in the subscript, but never mind that. Writing it that way doesn't do anything since you didn't actually take derivatives. Put f_j = u_j + iv_j, (j = 1, ..., k) and write |f_1(z)|^2 +...|f_k(z)|^2 in terms of the real and complex parts of each f_j. Then take two partial derivatives with respect to x and call that (1).

 

[FanofAFan]

Ok so i got |f₁(z)|² = f₁(z)f₁(z(bar)) = (x₁+iy₁)(x₁-iy₁) + ... + (x_n+iy_n)(x_n-iy_n)

then separated it into real and imaginare parts
Re{(x₁)(x₁) + ...+ (x_n)(x_n)
partial x = 2(x₁)+ 2(x₂) +...+ 2(x_n)
or did you mean partial x = 2(x₁)(dx/dz)+ 2(x₂)(dx/dz) +...+ 2(x_n)(dx/dz)? or do i need to keep it (x+iy)(x-iy) and then do the partial?