Complex numbers have applications in many scientific areas, including
signal processing,
control theory,
electromagnetism,
fluid dynamics,
quantum mechanics,
cartography, and
vibration analysis. Some of these applications are described below. Complex conjugation is also employed in
inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the
network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the
maximum power transfer theorem is looked for.
Geometry Shapes Three
non-collinear points u, v, w in the plane determine the
shape of the triangle \{u, v, w\}. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as S(u, v, w) = \frac {u - w}{u - v}. The shape S of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an
affine transformation), corresponding to the intuitive notion of shape, and describing
similarity. Thus each triangle \{u, v, w\} is in a
similarity class of triangles with the same shape.
Fractal geometry The
Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location c where iterating the sequence f_c(z)=z^2+c does not
diverge when
iterated infinitely. Similarly,
Julia sets have the same rules, except where c remains constant.
Triangles Every triangle has a unique
Steiner inellipse – an
ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The
foci of a triangle's Steiner inellipse can be found as follows, according to
Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the
cubic equation (x-a)(x-b)(x-c)=0, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
Algebraic number theory . As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in \mathbb{C}.
A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called
algebraic numbers – they are a principal object of study in
algebraic number theory. Compared to \overline{\mathbb{Q}}, the algebraic closure of \mathbb{Q}, which also contains all algebraic numbers, \mathbb{C} has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of
field theory to the
number field containing
roots of unity, it can be shown that it is not possible to construct a regular
nonagon using only compass and straightedge – a purely geometric problem. Another example is the
Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify
sums of squares.
Analytic number theory Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the
Riemann zeta function is related to the distribution of
prime numbers.
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.
Dynamic equations In
differential equations, it is common to first find all complex roots of the
characteristic equation of a
linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in
difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .
Linear algebra Since \C is algebraically closed, any non-empty complex
square matrix has at least one (complex)
eigenvalue. By comparison, real matrices do not always have real eigenvalues, for example
rotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any
real eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of
eigendecomposition is a useful tool for computing matrix powers and
matrix exponentials. Complex numbers often generalize concepts originally conceived in the real numbers. For example, the
conjugate transpose generalizes the
transpose,
hermitian matrices generalize
symmetric matrices, and
unitary matrices generalize
orthogonal matrices.
In applied mathematics 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
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 of the corresponding is the
amplitude and the
argument is the
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 x(t) = \operatorname{Re} \{X( t ) \} and X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } 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,
compress, restore, and otherwise process
digital audio signals, still images, and
video signals. Another example, relevant to the two side bands of
amplitude modulation of AM radio, is: \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}
In physics Electromagnetism and electrical engineering In
electrical engineering, the
Fourier transform is used to analyze varying
electric currents and
voltages. The treatment of
resistors,
capacitors, and
inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the
impedance. This approach is called
phasor calculus. In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current. Because the voltage in an AC circuit is oscillating, it can be represented as V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ), To obtain the measurable quantity, the real part is taken: v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t. The complex-valued signal is called the
analytic representation of the real-valued, measurable signal .
Fluid dynamics In
fluid dynamics, complex functions are used to describe
potential flow in two dimensions.
Quantum mechanics The complex number field is intrinsic to the
mathematical formulations 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
Schrödinger equation and Heisenberg's
matrix mechanics – make use of complex numbers.
Relativity In
special relativity and
general relativity, some formulas for the metric on
spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is
used in an essential way in
quantum field theory.) Complex numbers are essential to
spinors, which are a generalization of the
tensors used in relativity. ==Characterizations, generalizations and related notions==