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 real-valued functions u(x,y) and v(x,y) are the two equations:
Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function f(x + iy) = u(x,y) + iv(x,y).