What Is the Key Step Missing in Deriving Wirtinger Derivatives?

[paraboloid]

Let $$\bar{z} = x+iy$$.
We are given that $$x = \frac{z+\bar{z}}{2}$$ & $$y = \frac{z-\bar{z}}{2i}$$.

We are trying to derive $$\partial F/\partial\bar{z} = 1/2(\partial F/ \partial x + i \partial F/ \partial y)$$, where F(x,y) is some function of two real variables.

Using the chain rule I get $$\partial F/\partial\bar{z} = \partial F/\partial x\cdot\partial x/\partial\bar{z} + \partial F/\partial y\cdot\partial y/\partial\bar{z}$$.
This is the point where I know something is going wrong.

I replace $$\partial x/\partial\bar{z}$$ with $$\partial \frac{z+\bar{z}}{2}/\partial\bar{z}$$, and the same for y with $$\frac{z-\bar{z}}{2i}$$.

Taking the partial derivatives $$\partial \frac{z+\bar{z}}{2}/\partial\bar{z}$$ & $$\partial \frac{z-\bar{z}}{2i}/\partial\bar{z}$$,
I get $$\partial F/\partial\bar{z} = \partial F/\partial x\cdot\frac{1}{2}-\partial F/\partial y\cdot\frac{1}{2i} = \frac{1}{2}(\partial F/\partial x - \partial F/i\partial y)$$.

What key step am I missing that's leading me to the wrong expression?

 

[fzero]

Let $$\bar{z} = x+iy$$.
We are given that $$x = \frac{z+\bar{z}}{2}$$ & $$y = \frac{z-\bar{z}}{2i}$$.

We are trying to derive $$\partial F/\partial\bar{z} = 1/2(\partial F/ \partial x + i \partial F/ \partial y)$$, where F(x,y) is some function of two real variables.

Using the chain rule I get $$\partial F/\partial\bar{z} = \partial F/\partial x\cdot\partial x/\partial\bar{z} + \partial F/\partial y\cdot\partial y/\partial\bar{z}$$.
This is the point where I know something is going wrong.

I replace $$\partial x/\partial\bar{z}$$ with $$\partial \frac{z+\bar{z}}{2}/\partial\bar{z}$$, and the same for y with $$\frac{z-\bar{z}}{2i}$$.

Taking the partial derivatives $$\partial \frac{z+\bar{z}}{2}/\partial\bar{z}$$ & $$\partial \frac{z-\bar{z}}{2i}/\partial\bar{z}$$,
I get $$\partial F/\partial\bar{z} = \partial F/\partial x\cdot\frac{1}{2}-\partial F/\partial y\cdot\frac{1}{2i} = \frac{1}{2}(\partial F/\partial x - \partial F/i\partial y)$$.

What key step am I missing that's leading me to the wrong expression?

$$\frac{1}{2}(\partial F/\partial x - \partial F/i\partial y)= \frac{1}{2}(\partial F/\partial x + i \partial F/\partial y),$$

so I think you derived the stated result. Incidentally, you have a typo at the top. With the definitions of x and y that you used, $$\bar{z} = x - i y.$$