Analyticity and Laplacian Operator in Complex Functions: A Domain D Study

[FanofAFan]

Homework Statement

Let f(z) be analytic on a domain D. Let \Delta = (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) Set W = |f(z)|² show that W\DeltaW = (W_x)²+(W_y)²

Homework Equations

The Attempt at a Solution

W = (U²+V²)
\DeltaW = (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) (U²+V₂)

also 4(U²+V²)[(V_x)²+(U_y)²]

 

[praharmitra]

Well, firstly kindly clean up your texing. The latex code for superscript is more concisely - ^{...} .

As to your question. Do the following.

1. Write z = x + i y,\ \bar{z} = x- i y. The rewrite \partial_x,\partial_y,\ \Delta in terms of z and \bar{z}. (using chain rule for partial differentials.)

2. Note that W = |f(z)|^2 = f(z)\bar{f(z)} = f(z)f(\bar{z})

Now solve.

 

[FanofAFan]

Sorry about the latex coding or whatever... Thanks!

 

[FanofAFan]

I'm confused... so would the \partial_x = (1 + i\partialy/\partialx)(1 - i\partialy/\partialx) is it just \textit{z}\overline{z}?

 

[praharmitra]

Remember now, z and \bar{z} are my independent variables.

\partial_x = \frac{\partial x}{\partial z}\partial_z+\frac{\partial x}{\partial \bar{z}}\partial_{\bar{z}} = \partial_z + \partial_\bar{z}

Similarly, you can do it for y.

 

[FanofAFan]

Remember now, z and \bar{z} are my independent variables.

\partial_x = \frac{\partial x}{\partial z}\partial_z+\frac{\partial x}{\partial \bar{z}}\partial_{\bar{z}} = \partial_z + \partial_\bar{z}

Similarly, you can do it for y.

so what's the difference between \partial_x and other one with x?

 

[praharmitra]

so what's the difference between \partial_x and \partialx?

Oh, I'm sorry if I didn't clarify my notation. It is standard to call

\frac{\partial}{\partial x} as \partial_x.

 

[FanofAFan]

ok so partial over the partial of x (x^2+y^2) = 2x, right?
so for what I'm doing the partial of x over the partial of z (x + iy) = (1)(partial of x over the partial of z) right?

 

[praharmitra]

ok so partial over the partial of x (x^2+y^2) = 2x, right?
so for what I'm doing the partial of x over the partial of z (x + iy) = (1)(partial of x over the partial of z) right?

yes, that's right. The reason I'm asking you to write everything in terms of z and \bar{z} is that it makes the calculations very very easy. (And it is very to useful to know the behavior of \partial_x, etc. in terms of z and \bar{z} for future use)

What I want you to prove is

\Delta = \partial_x^2+\partial_y^2 = 4\partial_z\partial_{\bar{z}}
\partial_x = \partial_z + \partial_{\bar{z}}
\partial_y = i \partial_z - i \partial_{\bar{z}}

Now W\Delta W = f(z)f(\bar{z})\Delta f(z)f(\bar{z}) = 4f(z)f(\bar{z})\partial_zf(z)\partial_{\bar{z}}f(\bar{z})

Now (\partial_x W)^2+(\partial_y W)^2 = ?? (show that it is equal to the above expression.)