Andy
are you suggesting that you don't understand?? 
Derivatives and the Cauchy-Riemann equations
Just as in real analysis, a "smooth" complex function w = f(z) may have a derivative at a particular point in its domain Ω. In fact, the definition of the derivative
is analogous to the real case, with one very important difference. In real analysis, the limit can only be approached by moving along the one-dimensional number line. In complex analysis, the limit can be approached from any direction in the two-dimensional complex plane.
If this limit, the derivative, exists for every point z in Ω, then f(z) is said to be differentiable on Ω. It can be shown that any differentiable f(z) is analytic. This is a much more powerful result than the analogous theorem that can be proved for real-valued functions of real numbers. In the calculus of real numbers, we can construct a function f(x) that has a first derivative everywhere, but for which the second derivative does not exist at one or more points in the function's domain. But in the complex plane, if a function f(z) is differentiable in a neighborhood it must also be infinitely differentiable in that neighborhood.
By applying the methods of vector calculus to compute the partial derivatives of the two real functions u(x, y) and v(x, y) into which f(z) can be decomposed, and by considering two paths leading to a point z in Ω, it can be shown that the derivative exists if and only if
Equating the real and imaginary parts of these two expressions we obtain the traditional formulation of the Cauchy-Riemann Equations:
or, in another common notation,
By differentiating this system of two partial differential equations first with respect to x, and then with respect to y, we can easily show that
or, in another common notation,
In other words, the real and imaginary parts of a differentiable function of a complex variable are harmonic functions because they satisfy Laplace's equation.
I'm listening, continue
PS. I corrected your grammar (don't feel bad about it
