Editing Complex number (section)

===In applied mathematics===

====Control theory====
{{see also|Complex plane#Use in control theory}}

In [[control theory]], systems are often transformed from the [[time domain]] to the complex [[frequency domain]] using the [[Laplace transform]]. The system's [[zeros and poles]] are then analyzed in the ''complex plane''. The [[root locus]], [[Nyquist plot]], and [[Nichols plot]] techniques all make use of the complex plane.

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

* in the right half plane, it will be [[unstable]],
* all in the left half plane, it will be [[BIBO stability|stable]],
* on the imaginary axis, it will have [[marginal stability]].

If a system has zeros in the right half plane, it is a [[nonminimum phase]] system.

====Signal analysis====
Complex numbers are used in [[signal analysis]] and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a [[sine wave]] of a given [[frequency]], the absolute value {{math|{{!}}''z''{{!}}}} of the corresponding {{mvar|z}} is the [[amplitude]] and the [[Argument (complex analysis)|argument]] {{math|arg ''z''}} is the [[phase (waves)|phase]].

If [[Fourier analysis]] is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

<math display=block>x(t) = \operatorname{Re} \{X( t ) \} </math>

and

<math display=block>X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } </math>

where ω represents the [[angular frequency]] and the complex number ''A'' encodes the phase and amplitude as explained above.

This use is also extended into [[digital signal processing]] and [[digital image processing]], which use digital versions of Fourier analysis (and [[wavelet]] analysis) to transmit, [[Data compression|compress]], restore, and otherwise process [[Digital data|digital]] [[Sound|audio]] signals, still images, and [[video]] signals.

Another example, relevant to the two side bands of [[amplitude modulation]] of AM radio, is:

<math display=block>\begin{align}
  \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
    & = \operatorname{Re}\left(e^{i(\omega + \alpha)t} + e^{i(\omega - \alpha)t}\right) \\
    & = \operatorname{Re}\left(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \cdot e^{i\omega t}\right) \\
    & = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
    & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right).
\end{align}</math>