The original 2 statements linked seem false to me on their face. E.g. if f(z) = z = x+iy, then ∂f /∂x = 1 and ∂f/∂y = i everywhere, including along both axes. So it is hard for me to understand what the author means by his statements. This article uses the practice of manipulating notation in a suggestive way, without rigorously defining what is meant. Unless you find it entertaining, I suggest ignoring this confusing derivation and reading one more clear and correct, such as that of Henri Cartan, in Theory of functions of one and several complex variables, or Lang in his Complex Analysis, or for that matter, even Goursat's Course of mathematical analysis, tr. by Hedrick, volume II.
To me the clearest explanation of the Cauchy Riemann equations is one I have trouble finding a reference for at the moment, but which was also due to Cartan, in a book long out of print, until recently, called simply Diifferential Calculus, (now ... on Banach spaces, p.37). Namely regard f:C-->C as a map f:R^2-->R^2, assumed real differentiable, and take the derivative at a point p, which is of course a real linear mapping R^2-->R^2. If in fact that linear mapping is not only real linear, but also complex linear, which is to say it commutes with multiplication by i, or rotation counterclockwise through 90 degrees, then we say that f is complex differentiable, or holomorphic at p. That's it, the conditions expressing complex linearity of the derivative are the "Cauchy Riemann" equations.
To wit: A real linear map has a 2x2 matrix of real numbers, namely whose entries are the real partials ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y, and (you can check) this matrix defines a complex linear map (i.e. the matrix commutes with the matrix with rows [0 -1], [1 0], for a rotation), if and only if the rows have form [ a -b], [b a], i.e. iff ∂u/∂x = ∂v/∂y, and ∂u/∂y = -∂v/∂x.
I see I expressed it more briefly in 2006, in post #11 of this thread:
https://www.physicsforums.com/threads/cauchy-riemann-relations.140294/
Well maybe I am the last one to see this, but after re-reading the usual derivations of the C-R equations, including that of Riemann himself, the author seems to be trying to say that the limit of the difference quotient delta(f)/delta(z), should be independent of the direction in which delta(z) points. Riemann says that df/dz (he uses dw/dz) should be independent of dz. I am still unable to see how the author's words say this.
OK: I claim he is at least somewhat off the mark, and (some of) his statements are false. He asserts, as a consequence of his questionable and (I think) false statement that ∂f/∂y is "zero along the x axis", that therefore
∂f/∂z = (1/2)(∂u/∂x + i ∂v/∂x). But this is wrong. As everyone knows, for a holomorphic function,
∂f/∂z = ∂f/∂x, not 1/2 ∂f/∂x. E.g. just take f(z) = z again, so that ∂f/∂z = 1, but u=x so ∂u/∂x = 1, and v = y so ∂v/∂x = 0; hence (1/2)(∂u/∂x + i ∂v/∂x) = (1/2), and 1 ≠ 1/2.
In my opinion, this is going to make it hard to find a viewpoint from which his derivation works.
Although Cartan's explanation is the clearest theoretical one to me, the simplest derivation is the usual one, that we want df/dz to be a directional limit, independent of the direction dz points in. So when we approach in the x direction, i.e. when dz = dx, we get ∂f/∂x, and when we approach in the y direction, i.e. when dz = idy, we get ∂f/i∂y, so these must be equal, i.e. the C-R equations just say that
∂f/∂z = ∂f/∂x = (1/i)(∂f/∂ y) = -i∂f/∂y.
If you write out f = u+iv,
you get ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.I think this is what he is trying to say. Maybe someone can make sense of his approach, as suggested by FactChecker above, by working with differentials and setting dx = 0 along the y axis, etc..
maybe this? ∂f/∂z dz = ∂f/∂z (dx + idy) = ∂f/∂z dx + ∂f/∂z (idy). Now along the x axis, where dy = 0, and dz = dx, this says ∂f/∂z dx = ∂f/∂x dx, so we must have ∂f/∂z = ∂f/∂x. But along the y-axis where dx=0, and dz = idy, we have ∂f/∂z idy = ∂f/∂iy idy, so we must have ∂f/∂z = ∂f/i∂y. Thus we must have ∂f/∂z = df/∂x = ∂f/i∂y.
I still prefer Cartan's explanation: a real differentiable function has as derivative its best real linear approximation; a complex differentiable function has as derivative its best complex linear approximation; since every complex linear map is also real linear, thus every complex differentiable function is also real differentiable, with the same derivative, and a real differentiable function is complex differentiable if and only its unique best real linear approximation is also complex linear.
