http://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations
http://mathworld.wolfram.com/CauchyRiemannEquations.html
1. If z = a + bi, then a is called the real part of z, written a = R ez, and b is called the
imaginary part of z, written b = Imz. The absolute value or magnitude or modulus of z
is defined as (a2 +b2)1/2.
2. A complex number with magnitude 1 is said to be unimodular.
3. An argument of z (written arg z) is defined as the angle which the line segment from (0, 0)
to (a, b) makes with the positive real axis.
4. to a multiple of 2π.
If r is the magnitude of z and θ is an argument of z, we may write
z = r(cos θ + i sin θ)
5. The Cauchy–Riemann equations on a pair of realvalued functions u(x,y) and v(x,y) are the two equations:
 (1a)
and
 (1b)
Typically u and v are taken to be the real and imaginary parts respectively of a complexvalued function f(x + iy) = u(x,y) + iv(x,y).
6. If , then the CauchyRiemann equations become
(18)
 
7. If and satisfy the CauchyRiemann equations, they also satisfy Laplace's equation in two dimensions, since
(20)

8. An analytic signal is a signal with no negativefrequency components 9. A function is analytic if and only if it is equal to its Taylor series in someneighborhood of every point. 10. Any polynomial (real or complex) is an analytic function. This is because if a polynomial has degree n, any terms of degree larger than nin its Taylor series expansion will vanish, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series. 11.
Typical examples of functions that are not analytic are:
12. In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

1 comment:
A very well written post, illustrated with clear examples. Am sure must have taken you time to just write it.
Am surely impressed with your zeal to rediscover yourself and help others, to what we may have forgotten in our practical works :)
A small suggestion, probably you may replace the modulus in first point with the equation,
z = { (a^2) + (b^2) } ^ 0.5
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