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.