Control Theory
Incontrol theory, systems are often transformed from thetime domainto
thefrequency domainusing theLaplace transform. The
system'spolesandzerosare then analyzed in the complex plane. Theroot
locus,Nyquist plot, andNichols plottechniques all make use of the
complex plane.
In the root locus method, it is especially important whether
thepolesandzerosare in the left or right half planes, i.e. have real
part greater than or less than zero. If a system has poles that are
- in the right half plane, it will beunstable,
- all in the left half plane, it will bestable,
- on the imaginary axis, it will havemarginal stability.
If a system has zeros in the right half plane, it is anonminimum phasesystem.
Signal analysis
Complex numbers are used insignal 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 |z| of the corresponding z is the
amplitude and the argument arg (z) the phase.
If Fourier analysisis 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
ω f (t) = z
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
Improper integrals
In applied fields, complex numbers are often used to compute certain
real-valued improper integrals, by means of complex-valued functions.
Several methods exist to do this; see methods of contour integration.
