Partial derivative proof of complex numbers

Question

Given $z=x+iy$ and $\overline z=x-iy$, prove the following: $${\partial\over{\partial x}}={\partial\over{\partial z}}+{\partial\over{\partial{\overline z}}}$$ and $${\partial\over{\partial y}}=i\left({\partial\over{\partial z}}-{\partial\over{\partial{\overline z}}}\right)$$ I have no clue where to begin equations except that it may be related to Cauchy-Riemann equations. Any help as how to begin the proof and some hints to take note of would be of much help.

Answer 1

Let $u$ be an arbitrary continuous and differentiable function of $z$ and $\bar{z}$.

Also as $z = x+ iy$ and $\bar{z} = x-iy$,

$x = \frac{z+\bar{z}}{2}$ and $y = \frac{z-\bar{z}}{2i}$

So,

$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial z} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial z}$

$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial x}\frac{1}{2} + \frac{\partial u}{\partial y}\frac{1}{2i}$

or

$$\frac{\partial }{\partial z} = \frac{\partial }{\partial x}\frac{1}{2} - \frac{\partial }{\partial y}\frac{i}{2} \cdots(i)$$

Similarly,

$\frac{\partial u}{\partial \bar{ z}} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial \bar{z}} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial \bar{z}}$

$\frac{\partial u}{\partial \bar{z}} = \frac{\partial u}{\partial x}\frac{1}{2} + \frac{\partial u}{\partial y}\frac{-1}{2i}$

or $$\frac{\partial }{\partial \bar{z}} = \frac{\partial }{\partial x}\frac{1}{2} + \frac{\partial }{\partial y}\frac{i}{2}\cdots(ii)$$

Adding (i) and (ii),

$$\frac{\partial }{\partial z} + \frac{\partial }{\partial \bar{z}} = \frac{\partial }{\partial x}$$ or

$$\frac{\partial }{\partial x} = \frac{\partial }{\partial z} + \frac{\partial }{\partial \bar{z}}$$

Similarly by subtracting , we may find that

$$\frac{\partial }{\partial y} = i\big(\frac{\partial }{\partial z} - \frac{\partial }{\partial \bar{z}}\big)$$

Answer 2

$dz$ = $dx+ i dy$, $d \bar z = dx - i dy$

By definition, $\partial_z$ is such a combination of $\partial_x$ and $\partial_y$ that $dz(\partial_z) = 1$

Let's check if $\partial_z = (\partial_x - i\partial_y)/2$ fits.

$dx((\partial_x - i\partial_y)/2)+ idy((\partial_x - i\partial_y)/2) = \frac{1}{2} dx(\partial_x) + \frac{1}{2} idy(-i \partial_y) = 1$.

The same way you check that $\partial_{\bar{z}} = (\partial_x - i \partial_y) /2$

Dealing with the differentials is simple, because you can transform one differential into sum of others fast, but the vector fields are a bit harder to operate so I use duality between differential forms and vector fields. Since this duality is one-to-one, it's sufficient just to check that the equality holds.

Answer 3

$z=x+iy$ and $\overline z=x-iy$

Let $f=f(z,\overline z)$

Now ${\partial f\over{\partial x}}={\partial f\over{\partial z}} {\partial z\over{\partial{ x}}}+{\partial f\over{\partial{\overline z}}}{\partial {\overline z}\over{\partial{ x}}}={\partial f\over{\partial z}}(1)+{\partial f\over{\partial{\overline z}}}(1)=({\partial \over{\partial z}}+{\partial \over{\partial{\overline z}}})f$

$\implies{\partial\over{\partial x}}\equiv {\partial\over{\partial z}}+{\partial\over{\partial{\overline z}}}$

Similarly,

${\partial f\over{\partial y}}={\partial f\over{\partial z}} {\partial z\over{\partial{ y}}}+{\partial f\over{\partial{\overline z}}}{\partial {\overline z}\over{\partial{ y}}}={\partial f\over{\partial z}}(i)+{\partial f\over{\partial{\overline z}}}(-i)=i({\partial \over{\partial z}}-{\partial \over{\partial{\overline z}}})f$

$\implies{\partial\over{\partial y}}\equiv i({\partial\over{\partial z}}-{\partial\over{\partial{\overline z}}})$