1) On Wikipedia: "a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function."

2)

du/dx = dv/dy => True

-dv/dx => True

Thus, f'(z) = (y^2-x^2)/(x^2+y^2)^2 + (2 i x y)/(x^2+y^2)^2

Theorem 3.3.4If verifies the Cauchy-Riemann Formulas at and if the partial derivatives of and are continuous at , then is derivable at and . [source]