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We have a complex function $ w(z)=w(x+iy)$, and we can write $w(x,y)=u(x,y)+iv(x,y)$.

The derivative is $$\frac{dw}{dz}=\frac{1}{2}(\frac{\partial z}{\partial x}-i\frac{\partial w}{\partial y})$$ (right?)

There are lots of other ways to write this using the Cauchy-Riemann equation.

Now I want to understand the relation between the differentials $dz$, $dx$, $dy$.

I came across this equation: $dz\, d\bar{z}=dx\, dy$. But I cannot figure out why it would be true (or if it is?). I've tried substituting lots of stuff from the Cauchy-Riemann equations, but it doesn't seem to work out, and I don't really understand how the "algebra" with differentials works. Can anyone shed some light on this issue?

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:your question is exercise of complex variable and application (Churchill ) –  Maisam Hedyelloo Mar 9 '13 at 20:30

3 Answers 3

Hint:$$x=\frac{z+\bar z}{2}$$ and$$y=\frac{z-\bar z }{2i}$$ $$\frac{dw}{dz}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial z}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial z}$$$$\frac{dw}{dz}=(\frac{1}{2}\frac{\partial z}{\partial x})(\frac{-i}{2}\frac{\partial w}{\partial y})=\frac{1}{2}(\frac{\partial z}{\partial x}-i\frac{\partial w}{\partial y})$$

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shouldn't the second be a $y$ –  Dominic Michaelis Mar 9 '13 at 20:23

I think what is more of interest here is the relation

$$y \, dx - x \, dy = \frac{1}{2 i} (z \, d\bar{z} - \bar{z} \, dz)$$

Also,

$$\frac{d}{dz} = \frac{1}{2} \left (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right )$$

$$\frac{d}{d\bar{z}} = \frac{1}{2} \left (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right )$$

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Since none of the other posted answers have said anything about the 2-form in the question:

Because $z = x+iy$, the 1-form $dz$ can be written as $dz = dx +i\,dy$. Similarly, $d\bar z = d(x-iy) = dx - i\,dy$. Hence

\begin{align} d\bar z \wedge dz &= (dx - i\,dy) \wedge (dx + i\,dy) \\ &= dx \wedge dx - i\,dy\wedge dx + i\,dx\wedge dy + dy\wedge dy \\ &= 2i\,dx\wedge dy. \end{align}

which is useful when doing double integrals in complex coordinates. The area form $dx \wedge dy$ can thus be written as $\frac{1}{2i} d\bar z \wedge dz = \frac{i}{2} dz \wedge d\bar z$. One useful consequence is Cauchy's integral formula for $C^1$, not necessarily holomorphic, functions:

$$f(\zeta) = \frac{1}{2\pi i}\int_{\partial\Omega} \frac{f(z)}{z-\zeta}\,dz + \frac{1}{2\pi i} \iint_\Omega \frac{\frac{\partial f}{\partial \bar z}(z)}{z-\zeta}\,dz\wedge d\bar z.$$

(Note that if $f$ is holomorphic, then $\frac{\partial f}{\partial \bar z} = 0$, and the formula reduces to the familiar version of Cauchy's integral formula.)

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