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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

  (E.1)

so that

  (E.2)

and

  (E.3)

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

  (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*:

  (E.5)

These two numbers can be multiplied as

  (E.6)

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

 

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