Cauchy Reimann Equations Question?

[Buri]

I'd like to make sure of something. To begin with d/dx will denote partials. My text (Complex Analysis by Steine and Shakarchi) derives the equality df/dx = (1/i) df/dy. To derive this it considers the difference quotient by letting h be real and purely imaginary in another case. But it let's f(z) = f(x,y) where z = x + iy. So for real h we get df/dx and for purely imaginey h (1/i) df/dy. I was trying to cheek this with an example f(z) = z^2 and I "converted" this to it's correspondig real form f(x,y) = (x^2 - y^2, 2xy) and calculated the partials and then tested to see if it would work since f is Holomorphic. But it doesn't. Is this because the f in df/dx and the other is not the corresponding real function but rather the complex one? I converted the partials back to their complex forms and the equality did work. Is this the way you're supposed tenches these things? I guess I must use the cauchy reman equations instead with these partials, right?

 

[Buri]

Sorry for all the typos I'm on my phone lol

 

[Erland]

If f = u + i v (u, v real valued functions of x and y), then df/dx = du/dx + i dv/dx and df/fy = du/dy + i dv/dy (by definition). Using this, everything works fine, also for z^2 = (x^2 - y^2) + i 2xy.