∂x ∂v ∂y and ∂u

This article is accessible to undergraduate students who have completed a com-plex analysis course. The main purpose is to present the first steps in the theory of several complex variables and the Cauchy-Riemann theory. 1 Holomorphic functions of one variable In the identification of C with R2, a complex number is identified with a pair of real numbers, its real and imaginary part z = (x, y) = x+ iy. A continuous function f(z) = u(x, y) + iv(x, y) of one complex variable is called holomorphic ([GK]) if it fulfills the Cauchy-Riemann equations