Complex Numbers

Cuthbert Nyack

Notation


A complex number is written as z = a + jb where a is the real part of z and b is the imaginary part of z. j is the square root of -1 so j2 = - 1. A complex number can also be seen as a vector in a two dimensional space with axes Re z and Im z. In this space the vector will extend from the origin to the point (a, b). Every complex number a + jb has a complex conjugate given by a - jb.

Operations


For 2 complex numbers y = a + jb and z = c + jd the following holds.

y = z only if a = c and b = d. If a complex number is zero, then both its real and imaginary parts are zero.

Addition:- y + z = (a + c) + j(b + d)

Subtraction:- y - z = (a - c) + j(b - d)

Multiplication:- yz = (a + jb) (c + jd) = ac + jad + jbc + j2bd = (ac - bd) + j(ad + bc)

Reciprocal:- 1/(a + jb) = (a - jb)/ [(a + jb)(a - jb)] = (a - jb)/(a2 + b2)

Division:- y/z = (a + jb)/(c + jd) = [(a + jb)(c - jd)]/[(c + jd)(c - jd)] = [(ac + bd) + j(bc - ad)]/(c2 + d2)
This can also be written :- y/z = (ac + bd)/(c2 + d2) + j(bc - ad)/(c2 + d2)


Polar Form of Complex Numbers



A complex number z = a + jb can be written is polar form as z = r e jq where r2 = a2 + b2. r is called the magnitude or modulus or absolute value of z. and q is the phase or argument of z.


De Moivre's equation

( a + jb )n = (r ejq )n = rn (ejq )n = rn (ejnq ) = rn ( cos nq + j sin nq)
where r2 = a2 + b2
and q = tan-1(b/a)


Operations on complex numbers in polar form

Operations like multiplication, division, powers and roots are very easy if complex numbers are in polar form. Consider 3 complex numbers:-
z = r e jq
z1 = r1 e jq1
z2 = r2 e jq2

Multiplication:- z1z2 = r1r2 e j(q1 + q2)

Division:- z1/z2 = r1/r2 e j(q1 - q2)

Powers:- zn = rn e jnq

Roots:- z1/n = r1/n e j(q + 2mpi)/n

Logs:- ln z = ln r + j(q + 2mpi)


nth roots of unity

11/n = [e j 2pm]1/n = cos(2pm/n) + j sin(2pm/n)
The nth roots of unity are obtained for values of m from 0 to n - 1.

(-1)1/n = [e j (2m + 1)p]1/n = cos(p(2m + 1)/n) + j sin(p(2m + 1)/n)
The nth roots of -1 are obtained for values of m from 0 to n - 1.


Complex Numbers in Oscillations and Waves

For oscillations, the following relation is useful. e jw t = cos w t + j sin w t
cos w t = Re [e jw t]
sin w t = Im [e jw t]
For waves, the following is useful. e j(kx - w t) = cos(kx - w t) + j sin(kx - w t)


Functions of a Complex Variable

A function f(s) = 1/(s + a), where s is a complex number, is a function of a complex variable s or a complex function. Of particular interest in complex functions is the location of the poles(points at which the function goes to infinity) of the function. In f(s) there is only one pole at s = - a.
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Copyright 1996 Cuthbert A. Nyack.