67 Sentences With "complex conjugate" | Random Sentence Generator

The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Ylm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.

The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.

Every imaginary point belongs to exactly one real line, the line through the point and its complex conjugate.

Quaternionic representations are similar to real representations in that they are isomorphic to their complex conjugate representation. Here a real representation is taken to be a complex representation with an invariant real structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^2=+1. A representation which is isomorphic to its complex conjugate, but which is not a real representation, is sometimes called a pseudoreal representation.

For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation.

It may be that all the roots are real or instead there may be some that are complex numbers. In the latter case, all the complex roots come in complex conjugate pairs.

Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis. In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis.

In the other cases, the intersection of with the ombilic consists of two different pairs of complex conjugate points. As is a curve of degree two, its intersection with the plane at infinity consists of two points, possibly equal. The curve is thus a circle, if these two points are one of these two pairs of complex conjugate points on the ombilic. Each of these pairs defines a real line (passing through the points), which is the intersection of with the plane at infinity.

A map of the Riemann sphere onto itself is conformal if and only if it is a Möbius transformation. The complex conjugate of a Möbius transformation preserves angles, but reverses the orientation. For example, circle inversions.

In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. Preview available at Google books It follows from this (and the fundamental theorem of algebra), that if the degree of a real polynomial is odd, it must have at least one real root. That fact can also be proven by using the intermediate value theorem.

If there is no degeneracy they can only differ by a factor. In the time-dependent equation, complex conjugate waves move in opposite directions. If is one solution, then so is . The symmetry of complex conjugation is called time-reversal symmetry.

In the expressions above, A^\dagger is used as the symbol for the Hermitian conjugate (also called the conjugate transpose) of A, defined as applying both the complex conjugate and the transpose transformations to the operator A, in any order. The dagger (\dagger) is an old notation in mathematics, but is still widespread in quantum-mechanics. In mathematics (particularly linear algebra) the Hermitian conjugate of A is commonly written as A^\ast, but in quantum mechanics the asterisk (\ast) notation is sometimes used for the complex conjugate only, and not the combined conjugate transpose (Hermitian conjugate).

In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors. A real representation on a complex vector space is isomorphic to its complex conjugate representation, but the converse is not true: a representation which is isomorphic to its complex conjugate but which is not real is called a pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e.

Therefore, the absolute value of z, the argument of z, the real part of z and the imaginary part of z are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate formed by complex conjugation.

In mathematics, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation,. although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each of them is a complex representation.

The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory. It describes the collapse of the wave function as resulting from a time-symmetric transaction between a possibility wave from the source to the receiver (the wave function) and a possibility wave from the receiver to source (the complex conjugate of the wave function). This interpretation of quantum mechanics is unique in that it not only views the wave function as a real entity, but the complex conjugate of the wave function, which appears in the Born rule for calculating the expected value for an observable, as also real.

A function on complex affine space is holomorphic if its complex conjugate is Lie derived along the difference space V. This gives any complex affine space the structure of a complex manifold. Every affine function from A to the complex numbers is holomorphic. Hence, so is every polynomial in affine functions.

For example, in the representation theory of SL(2,C) these represent a pair of complex numbers i.e. spinors transforming according to the fundamental representation of SL(2,C) and the complex conjugate respectively. On the other hand, for the CG problem of SU(1,1), they transform according to two distinct SU(1,1) groups.

In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow. The algorithm finds the roots in complex conjugate pairs using only real arithmetic. See root-finding algorithm for other algorithms.

Consimilarity arises as a result of studying antilinear transformations referred to different bases. A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar to a real matrix and to a Hermitian matrix. There is a standard form for the consimilarity class, analogous to the Jordan normal form.

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner. stands for Darstellung, which means "representation" in German.

Bra–ket notation was designed to facilitate the formal manipulation of linear- algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, and denote arbitrary complex numbers, denotes the complex conjugate of , and denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

Therefore, Apollonius' problem has at most eight independent solutions (Figure 2). One way to avoid this double-counting is to consider only solution circles with non-negative radius. The two roots of any quadratic equation may be of three possible types: two different real numbers, two identical real numbers (i.e., a degenerate double root), or a pair of complex conjugate roots.

It can also be explained with reference to the sampling and Nyquist frequency. Take a window of N samples from an arbitrary real-valued signal at sampling rate fs . Taking the Fourier transform produces N complex coefficients. Of these coefficients only half are useful (the last N/2 being the complex conjugate of the first N/2 in reverse order, as this is a real valued signal).

If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola. If the points at infinity are the cyclic points and , the conic section is a circle. If the coefficients of a conic section are real, the points at infinity are either real or complex conjugate.

In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its complex conjugate and the curvature of the bundle and the torsion of the metric of the manifold. It is named after Salomon Bochner, Kunihiko Kodaira, and Shigeo Nakano.

Depending on the initial conditions, even with all roots real the iterates can experience a transitory tendency to go above and below the steady state value. But true cyclicity involves a permanent tendency to fluctuate, and this occurs if there is at least one pair of complex conjugate characteristic roots. This can be seen in the trigonometric form of their contribution to the solution equation, involving and .

The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair. M11 has two 5-dimensional irreducible representations over the field with 3 elements, related to the restrictions of 6-dimensional representations of the double cover of M12.

The complex square function is a twofold cover of the complex plane, such that each non-zero complex number has exactly two square roots. This map is related to parabolic coordinates. The absolute square of a complex number is the product involving its complex conjugate; it can also be expressed in terms of the complex modulus or absolute value, . It can be generalized to vectors as the complex dot product.

But it is important to note that a real variety may be a manifold and have singular points. For example the equation y^3 + 2 x^2 y - x^4 = 0 defines a real analytic manifold but has a singular point at the origin.Milnor, pp. 12–13 This may be explained by saying that the curve has two complex conjugate branches that cut the real branch at the origin.

This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as a quadratic or a cubic equation), then so is its conjugate.

Triangles with sides in ratio of 1/ form a closed spiral The Padovan sequence numbers can be written in terms of powers of the roots of the equation : x^3 -x -1 = 0.\, This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r.Richard Padovan, "Dom Hans Van Der Laan and the Plastic Number", pp. 181-193 in Nexus IV: Architecture and Mathematics, eds.

Music power has been making a comeback in recent years. See also Audio power. : Power specifications require the load impedance to be specified, and in some cases two figures will be given (for instance, the output power of a power amplifier for loudspeakers will be typically measured at 4 and 8 ohms). To deliver maximum power to the load, the impedance of the driver should be the complex conjugate of the impedance of the load.

The non-degenerate cylindrical sections are ellipses (or circles). When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric (i.e., the quadratic equation has real coefficients) can all be considered as a pair of lines, possibly coinciding. The empty set may be the line at infinity considered as a double line, a (real) point is the intersection of two complex conjugate lines and the other cases as previously mentioned.

The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis. In this example, one way to avoid these problems is to use the Q-variety Spec(Q[x1,x2,x3]/(x12+ x22+ 2x32- 2x1x3 - 2x2x3)), whose associated Q[i]-algebraic set is the union of the Q[i]-variety Spec(Q[i][x1,x2,x3]/(x1 + ix2 - (1+i)x3)) and its complex conjugate.

If the parabola intersects the -axis in two points, there are two real roots, which are the -coordinates of these two points (also called -intercept). If the parabola is tangent to the -axis, there is a double root, which is the -coordinate of the contact point between the graph and parabola. If the parabola does not intersect the -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.

In general, magneto-optic effects break time reversal symmetry locally (i.e. when only the propagation of light, and not the source of the magnetic field, is considered) as well as Lorentz reciprocity, which is a necessary condition to construct devices such as optical isolators (through which light passes in one direction but not the other). Two gyrotropic materials with reversed rotation directions of the two principal polarizations, corresponding to complex-conjugate ε tensors for lossless media, are called optical isomers.

If is a sphere, its intersection with the plane at infinity is the ombilic, and all plane sections are circles. If is a surface of revolution, its intersection with the ombilic consists of a pair of complex conjugate points (which are double points). A real plane contains these two points if and only if it is perpendicular to the axis of revolution. Thus the circular sections are the plane sections by a plane perpendicular to the axis, that have at least two real points.

In fact, if is a non-real root of , then its complex conjugate is also a root of . So, the product :(x-r)(x-s) = x^2-(r+s)x+rs =x^2+2ax+a^2+b^2 is a factor of that has real coefficients. This grouping of non-real factors may be continued until getting eventually a factorization with real factors that are polynomials of degrees one or two. For computing these real or complex factorizations, one has to know the roots of the polynomial.

The conic consists thus of two complex conjugate lines that intersect in the unique real point, (0,0), of the conic. The pencil of ellipses of equations ax^2+b(y^2-1)=0 degenerates, for a=0, b=1, into two parallel lines and, for a=1, b=0, into a double line. The pencil of circles of equations a(x^2+y^2-1) - bx =0 degenerates for a=0 into two lines, the line at infinity and the line of equation x=0.

An abstract CR structure on a real manifold M of real dimension n consists of a complex subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which has zero intersection with its complex conjugate. The CR codimension of the CR structure is k = n - 2 \dim L, where dim L is the complex dimension. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type.

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number. In a similar sense, one can define an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form . It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be interpreted geometrically.

For the time-independent equation, an additional feature of linearity follows: if two wave functions and are solutions to the time-independent equation with the same energy , then so is any linear combination: : \hat H (a\psi_1 + b \psi_2 ) = a \hat H \psi_1 + b \hat H \psi_2 = E (a \psi_1 + b\psi_2). Two different solutions with the same energy are called degenerate. In an arbitrary potential, if a wave function solves the time-independent equation, so does its complex conjugate, denoted . By taking linear combinations, the real and imaginary parts of are each solutions.

Effect of pole location on a second order system's natural frequency and damping ratio. This pole's complex conjugate (which necessarily exists since this pole has a nonzero imaginary component) is not shown. In addition to determining the stability of the system, the root locus can be used to design the damping ratio (ζ) and natural frequency (ωn) of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin.

In mathematics, dianalytic manifolds are possibly non-orientable generalizations of complex analytic manifolds. A dianalytic structure on a manifold is given by an atlas of charts such that the transition maps are either complex analytic maps or complex conjugates of complex analytic maps. Every dianalytic manifold is given by the quotient of an analytic manifold (possibly non-connected) by a fixed-point-free involution changing the complex structure to its complex conjugate structure. Dianalytic manifolds were introduced by , and dianalytic manifolds of 1 complex dimension are sometimes called Klein surfaces.

In mathematics, a complex representation is a representation of a group (or that of Lie algebra) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real.

In the 802.16e OFDM mode, the initial estimated CFO is within [-2f_S,2f_S]. Besides this estimation, additional frequency offset of \pm 12f_S, \pm 8f_S, or \pm 4f_S, is possible given a CFO range of \pm 11f_S. In order to estimate this additional integer CFO, a matched filter matching the fractional CFO-compensated received signal against the modulated long preamble waveforms can be used. The coefficients of the matched filter are the complex conjugate of the long preamble and they are modulated by a sinusoidal wave whose frequency is a possible integer CFO mentioned above.

In the case of real coefficients, it is positive if and only if the polynomial has two distinct real roots. Similarly for a cubic polynomial, the discriminant is zero if and only if the polynomial has a multiple root. In the case of real coefficients, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two distinct complex conjugate roots. More generally, the discriminant of a polynomial of positive degree is zero if and only if the polynomial has a multiple root.

The name ‘ptychography’ was coined by Hegerl and Hoppe in 1970 to describe a solution to the crystallographic phase problem first suggested by Hoppe in 1969. The idea required the specimen to be highly ordered (a crystal) and to be illuminated by a precisely engineered wave so that only two pairs of diffraction peaks interfere with one another at a time. A shift in the illumination changes the interference condition (via the Fourier shift theorem). The two measurements can be used to solve for the relative phase between the two diffraction peaks by breaking a complex-conjugate ambiguity that would otherwise exist.

The first case corresponds to the usual situation; each pair of roots corresponds to a pair of solutions that are related by circle inversion, as described below (Figure 6). In the second case, both roots are identical, corresponding to a solution circle that transforms into itself under inversion. In this case, one of the given circles is itself a solution to the Apollonius problem, and the number of distinct solutions is reduced by one. The third case of complex conjugate radii does not correspond to a geometrically possible solution for Apollonius' problem, since a solution circle cannot have an imaginary radius; therefore, the number of solutions is reduced by two.

In mathematics, a trigonometric number is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of , or the sine or cosine of a rational number of degrees. One of the simplest examples is \cos \frac \pi 4=\frac \sqrt 2 2. A real number different from is a trigonometric number if and only if it is the real part of a root of unity (see Niven's theorem). Thus every trigonometric number is half the sum of two complex conjugate roots of unity.

For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate character. Consider a character ψ of the idele class group, taken to be a map into the unit circle which is 1 on principal ideles and on an exceptional finite set S containing all infinite places. Then ψ generates a character χ of the ideal group IS, the free abelian group on the prime ideals not in S.Heilbronn (1967) p.

If this voltage exceeds the dielectric breakdown strength of the insulating material of the line then an arc will occur. This in turn can cause a reactive pulse of high voltage that can destroy the transmitter's final output stage. In RF systems, typical values for line and termination impedance are 50 Ω and 75 Ω. To maximise power transmission for radio frequency power systems the circuits should be complex conjugate matched throughout the power chain, from the transmitter output, through the transmission line (a balanced pair, a coaxial cable, or a waveguide), to the antenna system, which consists of an impedance matching device and the radiating element(s).

The fixed points of τ correspond to the boundary points of Σ/τ. The surface Σ is called an "analytic double" of Σ/τ. The Klein surfaces form a category; a morphism from the Klein surface X to the Klein surface Y is a differentiable map f:X→Y which on each coordinate patch is either holomorphic or the complex conjugate of a holomorphic map and furthermore maps the boundary of X to the boundary of Y. There is a one-to-one correspondence between smooth projective algebraic curves over the reals (up to isomorphism) and compact connected Klein surfaces (up to equivalence). The real points of the curve correspond to the boundary points of the Klein surface.

A square matrix A that is equal to its transpose, that is, A = A, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −A, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.

He was the first who discovered the rules for summing the > powers of the roots of any equation. In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details. The cubic was first partly solved by the 15–16th-century Italian mathematician Scipione del Ferro, who did not however publish his results; this method, though, only solved one type of cubic equation.

Nigel Hitchin introduced the notion of a generalized almost complex structure on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM. A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle. An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the Lie bracket.

The definition of the complex molecular multipole moment given above is the complex conjugate of the definition given in this article, which follows the definition of the standard textbook on classical electrodynamics by Jackson, except for the normalization. Moreover, in the classical definition of Jackson the equivalent of the N-particle quantum mechanical expectation value is an integral over a one-particle charge distribution. Remember that in the case of a one-particle quantum mechanical system the expectation value is nothing but an integral over the charge distribution (modulus of wavefunction squared), so that the definition of this article is a quantum mechanical N-particle generalization of Jackson's definition. The definition in this article agrees with, among others, the one of Fano and RacahU.

This happens as follows: Due to CP-symmetry violating electroweak interactions, some amplitudes involving quarks are not equal to the corresponding amplitudes involving anti-quarks, but rather have opposite phase (see CKM matrix and Kaon); since time reversal takes an amplitude to its complex conjugate, CPT-symmetry is conserved. Though some of their amplitudes have opposite phases, both quarks and anti-quarks have positive energy, and hence acquire the same phase as they move in space-time. This phase also depends on their mass, which is identical but depends both on flavor and on the Higgs VEV which changes along the domain wall. Thus certain sums of amplitudes for quarks have different absolute values compared to those of anti-quarks.

Most quantum option pricing research typically focuses on the quantization of the classical Black–Scholes–Merton equation from the perspective of continuous equations like the Schrödinger equation. Haven builds on the work of Chen and others, but considers the market from the perspective of the Schrödinger equation. The key message in Haven's work is that the Black–Scholes–Merton equation is really a special case of the Schrödinger equation where markets are assumed to be efficient. The Schrödinger-based equation that Haven derives has a parameter ħ (not to be confused with the complex conjugate of h) that represents the amount of arbitrage that is present in the market resulting from a variety of sources including non-infinitely fast price changes, non- infinitely fast information dissemination and unequal wealth among traders.

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots. The square root of the product of the discriminant by (and possibly also by the square of a rational number) appears in the formulas for the roots of a cubic polynomial. If the polynomial is irreducible and its coefficients are rational numbers (or belong to a number field), then the discriminant is a square of a rational number (or a number from the number field) if and only if the Galois group of the cubic equation is the cyclic group of order three.

An important principle used in crystal radio design to transfer maximum power to the earphone is impedance matching. The maximum power is transferred from one part of a circuit to another when the impedance of one circuit is the complex conjugate of that of the other; this implies that the two circuits should have equal resistance. However, in crystal sets, the impedance of the antenna-ground system (around 10-200 ohms) is usually lower than the impedance of the receiver's tuned circuit (thousands of ohms at resonance), and also varies depending on the quality of the ground attachment, length of the antenna, and the frequency to which the receiver is tuned. Therefore, in improved receiver circuits, in order to match the antenna impedance to the receiver's impedance, the antenna was connected across only a portion of the tuning coil's turns.

The degenerate conic is either: a point, when the plane intersects the cone only at the apex; a straight line, when the plane is tangent to the cone (it contains exactly one generator of the cone); or a pair of intersecting lines (two generators of the cone). These correspond respectively to the limiting forms of an ellipse, parabola, and a hyperbola. If a conic in the Euclidean plane is being defined by the zeros of a quadratic equation (that is, as a quadric), then the degenerate conics are: the empty set, a point, or a pair of lines which may be parallel, intersect at a point, or coincide. The empty set case may correspond either to a pair of complex conjugate parallel lines such as with the equation x^2+1=0, or to an imaginary ellipse, such as with the equation x^2 +y^2+1=0.

In the solution equation :x_t = c_1\lambda_1^t +\cdots + c_n\lambda_n^t, a term with real characteristic roots converges to 0 as grows indefinitely large if the absolute value of the characteristic root is less than 1. If the absolute value equals 1, the term will stay constant as grows if the root is +1 but will fluctuate between two values if the root is −1. If the absolute value of the root is greater than 1 the term will become larger and larger over time. A pair of terms with complex conjugate characteristic roots will converge to 0 with dampening fluctuations if the absolute value of the modulus of the roots is less than 1; if the modulus equals 1 then constant amplitude fluctuations in the combined terms will persist; and if the modulus is greater than 1, the combined terms will show fluctuations of ever-increasing magnitude.

Then one orientation-reversing isometry g of is given by , where denotes the complex conjugate of z. These facts imply that the mapping given by is an orientation-reversing isometry of that generates an infinite cyclic group G of isometries. (It can be expressed as , and its square is the isometry , which can be expressed as .) The quotient of the action of this group can easily be seen to be topologically a Möbius band. But it is also easy to verify that it is complete and non-compact, with constant negative curvature equal to −1. The group of isometries of this Möbius band is 1-dimensional and is isomorphic to the special orthogonal group SO(2). (Constant) zero curvature: This may also be constructed as a complete surface, by starting with portion of the plane R2 defined by and identifying with for all x in R (the reals).

If X is a Klein surface, a function f:X→Cu{∞} is called meromorphic if, on each coordinate patch, f or its complex conjugate is meromorphic in the ordinary sense, and if f takes only real values (or ∞) on the boundary of X. Given a connected Klein surface X, the set of meromorphic functions defined on X form a field M(X), an algebraic function field in one variable over R. M is a contravariant functor and yields a duality (contravariant equivalence) between the category of compact connected Klein surfaces (with non-constant morphisms) and the category of function fields in one variable over the reals. One can classify the compact connected Klein surfaces X up to homeomorphism (not up to equivalence!) by specifying three numbers (g, k, a): the genus g of the analytic double Σ, the number k of connected components of the boundary of X , and the number a, defined by a=0 if X is orientable and a=1 otherwise. We always have k ≤ g+1. The Euler characteristic of X equals 1-g.