The closest link I found from (1) to (2) uses differential geometry and in particular the exterior algebra. Using the exterior derivative df = (df/dx) dx + (df

The closest link I found from (1) to (2) uses differential geometry and in particular the exterior algebra. Using the exterior derivative df = (df/dx) dx + (df/dy) dy. it is possibly to define (identifying C with R^2) a function to be holomorphic when it can be written df = g (dx + idy) for some function g. From the properties of the exterior derivative you then get 0 = d(df) = d( g dz) = df' ^ dz, which implies that dg = h dz for some h, hence g is holomorphic as well. Defining the complex derivative of 'f' to be 'g' at least shows that any holomorphic function is infinitely complex differentiable. I haven't quite managed to figure out why this definition of the complex derivative coincides with the limit of f(w+z)/z as z->0 with z complex, but it should follow from the algebraic properties of C. Showing holomorphic functions to be analytic is somewhat trickier, but you can probably use that the above definition of holomorphicity and Stokes' theorem together imply Cauchy's integral theorem (every integral around a loop vanishes).