O'Reilly logo

Quantum Optics for Engineers by F.J. Duarte

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

i2=1

(E.1)

so that

ii=1

(E.2)

and

i(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*=aib

(E.5)

These two numbers can be multiplied as

cc*=(a+ib)(aib)=a2+b2

(E.6)

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

|c|

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required