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.
Residue theorem
The residue theorem in complex analysisis a powerful tool to evaluate
path integrals of meromorphic functions over closed curves and can
often be used to compute real integrals as well. It generalizes the
Cauchy and Cauchy's integral formula.
The statement is as follows. Suppose U is a simply connected open
subset of the complex plane C, a1,..., an are finitely many points of U
and f is a function which is defined and holomorphic on U\\{a1,...,an}.
If γ is a rectifiable curve in which doesn't meet any of the points ak
and whose start point equals its endpoint, then
Here, Res(f,ak) denotes the residue off at ak, and n(γ,ak) is the
winding number of the curve γ about the point ak. This winding number is
an integer which intuitively measures how often the curve γ winds
around the point ak; it is positive if γ moves in a counter clockwise
("mathematically positive") manner around ak and 0 if γ doesn't move
around ak at all.
In order to evaluate real integrals, the residue theorem is used in
the following manner: the integrand is extended to the complex plane and
its residues are computed (which is usually easy), and a part of the
real axis is extended to a closed curve by attaching a half-circle in
the upper or lower half-plane. The integral over this curve can then be
computed using the residue theorem. Often, the half-circle part of the
integral will tend towards zero if it is large enough, leaving only the
real-axis part of the integral, the one we were originally interested
Quantum mechanics
The complex number field is relevant in the mathematical formulation
of quantum mechanics, where complex Hilbert spaces provide the context
for one such formulation that is convenient and perhaps most standard.
The original foundation formulas of quantum mechanics - the Schrodinger
equation and Heisenberg's matrix mechanics - make use of complex
numbers.
The quantum theory provides a quantitative explanation for two
types of phenomena that classical mechanics and classical
electrodynamics cannot account for:
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