Laplace operator under complex change of variables

[la0s]

Homework Statement

Let Δf= d^2f/dx^2+ d^g/dy^2 (laplace equation - Partial Derivatives) Show Δ(f(g(z))= Mod(g'(z))^2 * Δf(w,v) where g(z)=w(x,y)+v(x,y)i

Homework Equations

we propably need to use cauchy riemman equations: dw/dx = dv/dy and dw/dy = - dv/dx
and chain rule

The Attempt at a Solution

∆f(g(x,y)) = d^2 f/dw^2*((dw/dx)^2 + (dw/dy)^2) + d^2f/dv^2*((dv/dx)^2 + (dv/dy)^2) + 2*d^2f/dwdv*(dw/dx dv/dx + dw/dy dv/dy) + df/dw*(d^2w/dx^2 + d^2w/dy^2) + df/dv*(d^2v/dx^2 + d^2 v/dy^2).

If you use the C-R equations this reduces to the identity you stated

Someone gave me this solution but i can't understand where the term 2*d^2f/dwdv*(dw/dx dv/dx + dw/dy dv/dy) came from?

 

[marlon]

What's Mod ?

marlon

 

[la0s]

mod is the absolute value
ie lg'(z)l

 

[nidhi-g]

Hi

The term you cannot understand how to get does not contribute anything to the proof and with the CR equations it quals zero.

P.s are you a UCL maths student?