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The study of functions of a complex variable is known as
complex analysis and has enormous practical use in applied mathematics as well as in other branches ofmathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros 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 especially important whether the poles and zeros are 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
If a system has zeros in the right half plane, it is a nonminimum phase system.
Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.
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 the real part of complex valued functions of the form
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
In electrical engineering, this is done for varying voltages and currents. The treatment of
resistors, capacitors and inductors can then be unified by introducing imaginaryfrequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents.)
In applied fields, the use of complex analysis is 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.
The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.
In Special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.
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