Quantum Optics for Engineers

Appendix E: Complex Numbers

E.1  Complex Numbers

Here, we offer a brief and pragmatic introduction to complex numbers and some well-known trigonometric identities based on complex numbers.

The imaginary part of a complex number is represented by i. The number i has the basic property

$i^2=-1$

(E.1)

so that

$i\cdot i=-1$

(E.2)

and

$i\cdot (-i)=+1$

(E.3)

A complex number has a real and an imaginary part denoted by i. A complex number c is defined as

$c=a+ib$

(E.4)

where a and b are real. This complex number is depicted in Figure E.1. The complex conjugate of this number c is denoted by c*:

$c*=a-ib$

(E.5)

These two numbers can be multiplied as

$cc*=(a+ib)(a-ib)=a^2+b^2$

(E.6)

and the magnitude of c = a + ib is denoted by |c|:

$|c|$