Antenna Theory and Applications

Appendix C

Complex Analysis

The use of complex variables in solving problems in the applied sciences, as in electrical engineering, appears to be a very valuable tool. Especially when dealing with sinusoidal excitations, the introduction of complex variables will simplify the solution process. Before this simplification is fully appreciated though, we have to deal first with the somewhat awkward concept of complex numbers.

C.1 Complex Numbers

We are all familiar with the real numbers and the permitted and non-permitted operations on real numbers. So, it is, for example, permitted to calculate the square root of the number 3.79 (), but the square root of −4 does not exist.

The complex numbers allow for the last square root to exist, through the introduction of so-called imaginary numbers alongside the real numbers. Any complex number consists of a real part and an imaginary part and is generally denoted as

C.1

where c is a complex number, a is the real part of the complex number

C.2

and b is the imaginary part of the complex number

C.3

j is the imaginary unit1 that exhibits the special ...