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

${i}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-1$ |
(E.1) |

so that

$i\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-1$ |
(E.2) |

and

$i\cdot (-i)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}+1$ |
(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*|:

$|c|$ |

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