Complex Numbers and Functions
The real number domain can be broadened up to the complex numbers, for instance the square root of negative real number is not defined in the real number domain. The complex number c can be represented as the vector of ordered pair (a,b) having a corresponds to pure real number (a,0) and b corresponds to pure imaginary number (0,b). The complex numbers are often represented in a following way
c = Re c + iIm c = a + ib

Fig. 1: Complex numbers

One can see from Figure 1 that whether you know r (=point P vector) and angle j it's possible to unambiguously determine c (complex number value). The Euler Formula:
c = a + ib = r (cos j + i sin j) = r · eij
where r² = a² + b² and tan j = b/a.
Important complex number rules:

1. a. Addition (subtraction is analogous having minus sign)
   c1 + c2 = (a1 + a2, b1 + b2) = (a1 + a2) + i(b1 + b2)
2. b. Multiplication
   c1 · c2 = r1eij1 − r2eij2 = (r1 · r2) ei(j1 + j2)
3. c. Division
   c1/c2 = r1/r2 ei(j1 − j2)
4. Exponentiation
   cn = rn eijn
5. Complex-conjugate number c*
   c* = a − ib = re−ij
6. Absolute value
   Ö(|c|²) = Ö(cc*) = Ö(reij re−ij) = r
7. Polar representation
   i = eiπ/2 with i² = −1

The complex numbers representation is applied for instance when describing phase shifts alterations between alternate currents and voltages (complex resistance = impedance) or between electric alternate field and induced polarization (complex dielectric constant). The complex numbers are also used in description of periodical processes with e-functions serving Euler formula for real and imaginary amplitudes recalculation.