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$$\Delta$$W = (W_x)²+(W_y)²

Homework Equations

The Attempt at a Solution

W = (U²+V²)
$$\Delta$$W = ($$\frac{\partial^2}{\partial x^2}$$ + $$\frac{\partial^2}{\partial y^2}$$) (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$$\partial$$y/$$\partial$$x)(1 - i$$\partial$$y/$$\partial$$x) 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 $$\partial$$x?

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.)