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In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied casesSplit-complex number (4,177 words) [view diff] exact match in snippet view article find links to article
independent quantity. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined byComplex quadratic polynomial (2,947 words) [view diff] exact match in snippet view article find links to article
unicritical polynomial, i.e. it has one finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity andOrbit trap (439 words) [view diff] exact match in snippet view article find links to article
typically used to colour two dimensional fractals representing the complex plane. A point-based orbit trap colours a point based upon how close a function'sPoint at infinity (779 words) [view diff] exact match in snippet view article find links to article
can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projectiveError function (6,897 words) [view diff] exact match in snippet view article find links to article
( − t 2 ) {\displaystyle \exp(-t^{2})} is holomorphic on the whole complex plane C {\displaystyle \mathbb {C} } . In many applications, the functionBloch space (136 words) [view diff] exact match in snippet view article find links to article
space of holomorphic functions f defined on the open unit disc D in the complex plane, such that the function ( 1 − | z | 2 ) | f ′ ( z ) | {\displaystyleUniform boundedness (352 words) [view diff] no match in snippet view article find links to article
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant isGamma function (13,517 words) [view diff] exact match in snippet view article find links to article
d}}t,\ \qquad \Re (z)>0\,.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphicAiry function (4,032 words) [view diff] exact match in snippet view article find links to article
... As explained below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic behaviour of the Airy functionsUnit hyperbola (1,507 words) [view diff] exact match in snippet view article find links to article
this unit hyperbola group is not compact. Similar to the ordinary complex plane, a point not on the diagonals has a polar decomposition using the parametrizationPhragmén–Lindelöf principle (1,991 words) [view diff] exact match in snippet view article find links to article
maximum modulus principle cannot be applied to an unbounded region of the complex plane. As a concrete example, let us examine the behavior of the holomorphicHyperbolic functions (4,824 words) [view diff] exact match in snippet view article find links to article
result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendentalScorer's function (510 words) [view diff] exact match in snippet view article find links to article
the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D Plot of the Scorer function Hi(z) in the complex planeSubharmonic function (1,833 words) [view diff] exact match in snippet view article find links to article
especially for the study of Hp when 0 < p < 1. In the context of the complex plane, the connection to the convex functions can be realized as well by theFundamental theorem of algebra (7,610 words) [view diff] exact match in snippet view article find links to article
|p(z)| on the whole complex plane is achieved at z0. If |p(z0)| > 0, then 1/p is a bounded holomorphic function in the entire complex plane since, for eachFunctional equation (1,632 words) [view diff] exact match in snippet view article find links to article
the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem)Nevanlinna theory (2,603 words) [view diff] exact match in snippet view article find links to article
functions of one complex variable defined in a disc |z| ≤ R or in the whole complex plane (R = ∞). Subsequent generalizations extended Nevanlinna theory to algebroidSchwarz alternating method (980 words) [view diff] exact match in snippet view article find links to article
the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz describedNevanlinna theory (2,603 words) [view diff] exact match in snippet view article find links to article
functions of one complex variable defined in a disc |z| ≤ R or in the whole complex plane (R = ∞). Subsequent generalizations extended Nevanlinna theory to algebroidOpen mapping theorem (247 words) [view diff] exact match in snippet view article find links to article
a non-constant holomorphic function on a connected open set in the complex plane is an open mapping Open mapping theorem (topological groups), statesTrigonometric integral (3,289 words) [view diff] exact match in snippet view article find links to article
Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be takenNyquist stability criterion (3,425 words) [view diff] exact match in snippet view article find links to article
1+G(s)} in the right-half complex plane minus the number of poles of 1 + G ( s ) {\displaystyle 1+G(s)} in the right-half complex plane. If instead, the contourResolvent formalism (866 words) [view diff] exact match in snippet view article find links to article
exists a simple closed curve C λ {\displaystyle C_{\lambda }} in the complex plane that separates λ from the rest of the spectrum of A. Then the residueEuler product (2,219 words) [view diff] exact match in snippet view article find links to article
proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. In general, ifLine bundle (1,885 words) [view diff] exact match in snippet view article find links to article
negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopyPainlevé transcendents (2,779 words) [view diff] exact match in snippet view article find links to article
certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles)Monodromy (1,483 words) [view diff] exact match in snippet view article find links to article
is an analytic function F(z) in some open subset E of the punctured complex plane C ∖ { 0 } {\displaystyle \mathbb {C} \backslash \{0\}} may be continuedFatou conjecture (60 words) [view diff] exact match in snippet view article find links to article
after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. ŚwiątekPrime omega function (4,153 words) [view diff] no match in snippet view article find links to article
In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factorsStokes phenomenon (1,323 words) [view diff] exact match in snippet view article find links to article
asymptotic behavior of functions can differ in different regions of the complex plane. This seemingly gives rise to a paradox when looking at the asymptoticPosition operator (2,610 words) [view diff] exact match in snippet view article find links to article
function of the position line – defined from the real line into the complex plane by x : R → C : x ↦ x . {\displaystyle \mathrm {x} :\mathbb {R} \to \mathbbMethod of steepest descent (5,057 words) [view diff] exact match in snippet view article find links to article
approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the directionWeierstrass–Enneper parameterization (1,815 words) [view diff] exact match in snippet view article find links to article
{\displaystyle f} and g {\displaystyle g} be functions on either the entire complex plane or the unit disk, where g {\displaystyle g} is meromorphic and f {\displaystyleSimply connected space (1,330 words) [view diff] exact match in snippet view article find links to article
continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is C ∖ { 0 } , {\displaystyle \mathbbInfinity (6,108 words) [view diff] exact match in snippet view article find links to article
\infty } can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resultingPlanar Riemann surface (5,224 words) [view diff] exact match in snippet view article find links to article
surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed. A closed 1-form ω is exactLehmer–Schur algorithm (1,787 words) [view diff] exact match in snippet view article find links to article
enclosing roots like in the one-dimensional bisection method to the complex plane. It uses the Schur-Cohn test to test increasingly smaller disks forArakelyan's theorem (178 words) [view diff] exact match in snippet view article find links to article
of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset. Let Ω be an open subsetGoogle matrix (1,796 words) [view diff] exact match in snippet view article find links to article
eigenvalues λ i {\displaystyle \lambda _{i}} of Google matrices in the complex plane at α = 1 {\displaystyle \alpha =1} for dictionary networks: Roget (ABiholomorphism (557 words) [view diff] exact match in snippet view article find links to article
{\displaystyle n=1,} every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (this is the Riemann mapping theorem)Euclidean plane isometry (3,411 words) [view diff] no match in snippet view article find links to article
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometricalDirected infinity (181 words) [view diff] exact match in snippet view article find links to article
A directed infinity is a type of infinity in the complex plane that has a defined complex argument θ but an infinite absolute value r. For example, theFixed-point index (341 words) [view diff] exact match in snippet view article find links to article
setting of complex analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphicBrown measure (326 words) [view diff] exact match in snippet view article find links to article
measure of an operator in a finite factor is a probability measure on the complex plane which may be viewed as an analog of the spectral counting measure (basedPoincaré metric (2,101 words) [view diff] exact match in snippet view article find links to article
expressed. These various forms are reviewed below. A metric on the complex plane may be generally expressed in the form d s 2 = λ 2 ( z , z ¯ ) d z dDirichlet eta function (3,706 words) [view diff] exact match in snippet view article find links to article
well as another means to extend the definition of eta to the entire complex plane. The zeros of the eta function include all the zeros of the zeta function:Continued fraction (8,710 words) [view diff] exact match in snippet view article find links to article
continued fractions that are rapidly convergent almost everywhere in the complex plane. The long continued fraction expression displayed in the introductionFunctional-theoretic algebra (996 words) [view diff] no match in snippet view article find links to article
Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionalsBackward Euler method (907 words) [view diff] exact match in snippet view article find links to article
in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. This includes the whole left half of the complex plane, makingDragon curve (1,569 words) [view diff] exact match in snippet view article find links to article
also the limit set of the following iterated function system in the complex plane: f 1 ( z ) = ( 1 + i ) z 2 {\displaystyle f_{1}(z)={\frac {(1+i)z}{2}}}Operator product expansion (1,013 words) [view diff] exact match in snippet view article find links to article
where increasing time has been mapped to some increasing radius on the complex plane. Normal ordering of creation operators is useful when working in theNikolai Georgievich Makarov (773 words) [view diff] exact match in snippet view article find links to article
harmonic measure in the complex plane. Makarov's theorem states that: Let Ω be a simply connected domain in the complex plane. Suppose that ∂Ω (the boundaryJacobi elliptic functions (13,085 words) [view diff] exact match in snippet view article find links to article
in the m {\displaystyle m} -plane remain to be investigated. In the complex plane of the argument u {\displaystyle u} , the twelve functions form a repeatingSelberg's zeta function conjecture (591 words) [view diff] exact match in snippet view article find links to article
known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on thisRigidity (mathematics) (753 words) [view diff] exact match in snippet view article
derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single pointConformal welding (1,123 words) [view diff] exact match in snippet view article find links to article
holomorphic maps f, g of the unit disk and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domainsExponential sheaf sequence (485 words) [view diff] exact match in snippet view article find links to article
sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g ofPositive-real function (1,012 words) [view diff] exact match in snippet view article find links to article
it has a positive real part and is analytic in the right half of the complex plane and takes on real values on the real axis. In symbols the definitionGeneralized Riemann hypothesis (1,318 words) [view diff] exact match in snippet view article find links to article
(only when χ {\displaystyle \chi } is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every DirichletInfinite product (1,456 words) [view diff] exact match in snippet view article find links to article
function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, eachSchwarz's list (904 words) [view diff] exact match in snippet view article find links to article
of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that casesHesse configuration (849 words) [view diff] exact match in snippet view article find links to article
triples of inflection points. If a given set of nine points in the complex plane is the set of inflections of an elliptic curve C, it is also the setKelvin functions (1,602 words) [view diff] exact match in snippet view article find links to article
Plot of the Kelvin function ber(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3DElliptic function (2,442 words) [view diff] exact match in snippet view article find links to article
\Lambda } acting on C {\displaystyle \mathbb {C} } . Geometrically the complex plane is tiled with parallelograms. Everything that happens in one fundamentalMittag-Leffler distribution (679 words) [view diff] exact match in snippet view article find links to article
}(Cz),} where the convergence stands for all z {\displaystyle z} in the complex plane if α ∈ ( 0 , 1 ] {\displaystyle \alpha \in (0,1]} , and all z {\displaystyleMcCullagh's parametrization of the Cauchy distributions (577 words) [view diff] exact match in snippet view article find links to article
on the sample space only if the parameter space is taken to be the complex plane. In other words, if the random variable Y has a Cauchy distributionHypergeometric function (6,914 words) [view diff] exact match in snippet view article find links to article
the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regularLambert W function (12,083 words) [view diff] exact match in snippet view article find links to article
are disjoint. The range of the entire multivalued function W is the complex plane. The image of the real axis is the union of the real axis and the quadratrixGaussian moat (481 words) [view diff] exact match in snippet view article find links to article
Unsolved problem in mathematics: In the complex plane, is it possible to "walk to infinity" in the Gaussian integers using the Gaussian primes as steppingClassification of Fatou components (614 words) [view diff] exact match in snippet view article find links to article
z ) {\displaystyle f={\frac {P(z)}{Q(z)}}} defined in the extended complex plane, and if it is a nonlinear function (degree > 1) d ( f ) = max ( degFourier (363 words) [view diff] exact match in snippet view article find links to article
series, a special case of the Z-transform around the unit circle in the complex plane Discrete Fourier transform (DFT), occasionally called the finite FourierLine spectral pairs (805 words) [view diff] exact match in snippet view article find links to article
can be shown that: The roots of P and Q lie on the unit circle in the complex plane. The roots of P alternate with those of Q as we travel around the circleIndex group (376 words) [view diff] exact match in snippet view article find links to article
Therefore, ΛL(H) is the trivial group. Let T denote the unit circle in the complex plane. The algebra C(T) of continuous functions from T to the complex numbersCircle bundle (993 words) [view diff] exact match in snippet view article find links to article
L} . For example, consider the analytification X {\displaystyle X} a complex plane curve Proj ( C [ x , y , z ] x n + y n + z n ) {\displaystyleAmplitwist (283 words) [view diff] exact match in snippet view article find links to article
amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number z {\displaystyle z} such thatFree particle (1,943 words) [view diff] exact match in snippet view article find links to article
or wave vector k, at angular frequency ω or energy E, is given by a complex plane wave: ψ ( r , t ) = A e i ( k ⋅ r − ω t ) = A e i ( p ⋅ r − E t ) /Singular integral operators of convolution type (12,876 words) [view diff] exact match in snippet view article find links to article
transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of theseBijection (2,508 words) [view diff] exact match in snippet view article find links to article
Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane. An injective non-surjective function (injectionSectrix of Maclaurin (2,647 words) [view diff] no match in snippet view article find links to article
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant ratesDe Branges space (390 words) [view diff] exact match in snippet view article find links to article
{\displaystyle |E(z)|>|E({\bar {z}})|} , for all z in the upper half of the complex plane C + = { z ∈ C ∣ Im ( z ) > 0 } {\displaystyle \mathbb {C} ^{+}=\{z\inAnalog signal processing (1,621 words) [view diff] exact match in snippet view article find links to article
transform of any system or signal because it is a transform into the complex plane instead of just the jω line like the Fourier transform. The major differenceUnitary matrix (1,336 words) [view diff] exact match in snippet view article find links to article
det ( U ) {\displaystyle \det(U)} will be on the unit circle of the complex plane. Its eigenspaces are orthogonal. U can be written as U = eiH, whereBrennan conjecture (272 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \mathbb {C} } with at least two boundary points in the extended complex plane. Let φ {\displaystyle \varphi } be a conformal map of W onto the openJ-invariant (4,723 words) [view diff] exact match in snippet view article find links to article
j has the property of mapping the fundamental region to the entire complex plane. Additionally two values τ,τ' ∈H produce the same elliptic curve iffSplitting circle method (2,184 words) [view diff] exact match in snippet view article find links to article
{\displaystyle p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}} for any region of the complex plane with a piecewise smooth boundary. Most of those factors will be trivialRotation matrix (15,046 words) [view diff] no match in snippet view article find links to article
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the conventionZ-transform (5,652 words) [view diff] exact match in snippet view article find links to article
inverse Z-transform. By integrating around a closed contour in the complex plane, the residues at the poles of the Z-transform function inside the ROCMonodromy theorem (1,331 words) [view diff] exact match in snippet view article find links to article
made precise below. Let U {\displaystyle U} be an open disk in the complex plane centered at a point P {\displaystyle P} and f : U → C {\displaystyleGenerator (mathematics) (645 words) [view diff] exact match in snippet view article
The 5th roots of unity in the complex plane under multiplication form a group of order 5. Each non-identity element by itself is a generator for the wholeComplex manifold (1,311 words) [view diff] exact match in snippet view article find links to article
complex manifolds are isomorphic to either: Δ, the unit disk in C C, the complex plane Ĉ, the Riemann sphere Note that there are inclusions between these asCartan's lemma (515 words) [view diff] exact match in snippet view article find links to article
variables: Let a1 < a2 < a3 < a4 and b1 < b2 and define rectangles in the complex plane C by K 1 = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 3 , b 1 < y 1 < b 2 }List of things named after Jacques Hadamard (268 words) [view diff] exact match in snippet view article find links to article
the maxima of holomorphic functions within concentric circles in the complex plane. Hadamard three-lines theorem, concerning the maxima of holomorphicDonald Sarason (1,025 words) [view diff] exact match in snippet view article find links to article
the modern development of function theory on the unit circle in the complex plane. In Sarason he showed that H ∞ + C {\displaystyle H^{\infty }+C} isAnger function (1,151 words) [view diff] exact match in snippet view article find links to article
Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3DArtin L-function (2,041 words) [view diff] exact match in snippet view article find links to article
a non-trivial irreducible representation ρ is analytic in the whole complex plane. This is known for one-dimensional representations, the L-functionsDouady–Earle extension (2,846 words) [view diff] exact match in snippet view article find links to article
Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension isComplex multiplication (2,071 words) [view diff] exact match in snippet view article find links to article
functions. More generally, consider the lattice Λ, an additive group in the complex plane, generated by ω 1 , ω 2 {\displaystyle \omega _{1},\omega _{2}} . ThenAhlfors theory (1,195 words) [view diff] exact match in snippet view article find links to article
coverings. Suppose now that Z is an open Riemann surface, for example the complex plane or the unit disc, and let Z be equipped with a conformal metric ds.Fibered knot (419 words) [view diff] exact match in snippet view article find links to article
complex algebraic geometry. For instance, each singular point of a complex plane curve can be described topologically as the cone on a fibered knot orBorel measure (1,336 words) [view diff] exact match in snippet view article find links to article
ISBN 0-09-113411-0. Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: CambridgeEI (493 words) [view diff] exact match in snippet view article find links to article
horse species Exponential integral, a special function defined on the complex plane given the symbol Ei Education Index, a United Nations measure of theWave packet (5,922 words) [view diff] exact match in snippet view article find links to article
1D Gaussian wave packet, shown in the complex plane, for a=2 and k=4Filled Julia set (899 words) [view diff] exact match in snippet view article find links to article
ISBN 978-0-387-15851-8. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-ReportH square (361 words) [view diff] exact match in snippet view article find links to article
and C + {\displaystyle \mathbb {C} ^{+}} is the right half of the complex plane. It is more; it is an isomorphism, in that it is invertible, and itMellin transform (4,679 words) [view diff] exact match in snippet view article find links to article
notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. ConditionsSmooth projective plane (1,679 words) [view diff] exact match in snippet view article find links to article
Theorem. A smooth 4-dimensional plane is isomorphic to the classical complex plane, or dim Aut P ≤ 6 {\displaystyle \dim \operatorname {Aut} {\mathcalLindsey–Fox algorithm (2,420 words) [view diff] exact match in snippet view article find links to article
Fourier transform) to very efficiently conduct a grid search in the complex plane to find accurate approximations to the N roots (zeros) of an Nth-degreeHypocycloid (1,342 words) [view diff] exact match in snippet view article find links to article
diagonal entries for a matrix in SU(3), are precisely the points in the complex plane lying inside a hypocycloid of three cusps (a deltoid). Likewise, summingKenneth Stewart Cole (1,378 words) [view diff] exact match in snippet view article find links to article
expression requires that the locus of the dielectric constant in the complex plane be a circular arc with end points on the axis of reals and center belowChandrasekhar's H-function (1,229 words) [view diff] no match in snippet view article find links to article
In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicistBirch and Swinnerton-Dyer conjecture (3,132 words) [view diff] exact match in snippet view article find links to article
the complex plane with Re(s) > 3/2. Helmut Hasse conjectured that L(E, s) could be extended by analytic continuation to the whole complex plane. ThisHadamard (disambiguation) (298 words) [view diff] exact match in snippet view article
complex plane; Hadamard three-circle theorem, a bound on the maximum modulus of complex analytic functions defined on an annulus in the complex plane;Andrew Browder (562 words) [view diff] case mismatch in snippet view article find links to article
several variables. In 2000 Browder published his article "Topology in the Complex Plane", which described the Brouwer fixed point theorem, the Jordan curveEuler's continued fraction formula (4,200 words) [view diff] exact match in snippet view article find links to article
series expansion that converges uniformly on every bounded domain in the complex plane. e x = 1 + ∑ n = 1 ∞ x n n ! = 1 + ∑ n = 1 ∞ ( ∏ i = 1 n x i ) {\displaystyleFunction field of an algebraic variety (683 words) [view diff] exact match in snippet view article find links to article
respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we sayHyperbolic orthogonality (1,181 words) [view diff] exact match in snippet view article find links to article
{\displaystyle z_{1}z_{2}^{*}+z_{1}^{*}z_{2}=0} entails perpendicularity in the complex plane, while w 1 w 2 ∗ + w 1 ∗ w 2 = 0 {\displaystyle w_{1}w_{2}^{*}+w_{1}^{*}w_{2}=0}Schwarzian derivative (6,967 words) [view diff] exact match in snippet view article find links to article
a=(Sf)(z_{0})} . After a translation, rotation, and scaling of the complex plane, ( M − 1 ∘ f ) ( z ) = {\displaystyle (M^{-1}\circ f)(z)={}} z + z 3Submersion (mathematics) (1,636 words) [view diff] exact match in snippet view article
{\displaystyle \mathbb {C} ,\mathbb {C} ^{2}} of the complex line and the complex plane. Note that we should actually remove the points t = 0 , 1 {\displaystyleInfinite compositions of analytic functions (4,981 words) [view diff] exact match in snippet view article find links to article
_{n+1}-\beta _{n}\right|<\infty } then Fn(z) → λ, a constant in the extended complex plane, for all z. The value of the infinite continued fraction a 1 b 1 + aDissipation factor (628 words) [view diff] exact match in snippet view article find links to article
When representing the electrical circuit parameters as vectors in a complex plane, known as phasors, a capacitor's dissipation factor is equal to theBicorn (410 words) [view diff] exact match in snippet view article find links to article
limaçon, has an ordinary double point at the origin, and two nodes in the complex plane, at x = ± i {\displaystyle x=\pm i} and z = 1 {\displaystyle z=1} .Multiple time dimensions (1,095 words) [view diff] exact match in snippet view article find links to article
imaginary time dimension, changing time from a real number line into a complex plane. Introducing it into Minkowski spacetime allows a generalization ofDiscrete spectrum (mathematics) (851 words) [view diff] exact match in snippet view article
\dots ).} For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrumNevanlinna–Pick interpolation (1,157 words) [view diff] exact match in snippet view article find links to article
interpolates a given set of data, where R is now an arbitrary region of the complex plane. M. B. Abrahamse showed that if the boundary of R consists of finitelyFundamental domain (1,005 words) [view diff] exact match in snippet view article find links to article
A lattice in the complex plane and its fundamental domain, with quotient a torus.Wrapped normal distribution (1,707 words) [view diff] exact match in snippet view article find links to article
estimator of the mean μ. Viewing the zn as a set of vectors in the complex plane, the R2 statistic is the square of the length of the averaged vector:Beltrami equation (10,901 words) [view diff] exact match in snippet view article find links to article
Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equationRamification (mathematics) (1,116 words) [view diff] exact match in snippet view article
analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface theoryNormal matrix (1,656 words) [view diff] exact match in snippet view article find links to article
all of its eigenvalues (its spectrum) lie on the unit circle of the complex plane. Proposition — A normal matrix is self-adjoint if and only if its spectrumXaoS (610 words) [view diff] case mismatch in snippet view article find links to article
Algorithms, archived from the original on 2014-03-28. Visualizations on the Complex Plane, archived from the original on 2006-10-17 Wikimedia Commons has mediaAlternating-direction implicit method (3,425 words) [view diff] exact match in snippet view article find links to article
{\displaystyle E} and F {\displaystyle F} are disjoint disks in the complex plane. When less is known about σ ( A ) {\displaystyle \sigma (A)} and σ (Alternating-direction implicit method (3,425 words) [view diff] exact match in snippet view article find links to article
{\displaystyle E} and F {\displaystyle F} are disjoint disks in the complex plane. When less is known about σ ( A ) {\displaystyle \sigma (A)} and σ (Mathematical visualization (757 words) [view diff] exact match in snippet view article find links to article
angles, dimensions and proportions In complex analysis, functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projectionLogarithmic integral function (1,101 words) [view diff] exact match in snippet view article find links to article
Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3DPolygamma function (2,380 words) [view diff] exact match in snippet view article find links to article
Plot of the digamma function, the first polygamma function, in the complex plane from −2−2i to 2+2i with colors created by Mathematica's function ComplexPlot3DSingular integral operators on closed curves (5,032 words) [view diff] exact match in snippet view article find links to article
Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case ofCartan's lemma (potential theory) (158 words) [view diff] exact match in snippet view article
found in Levin's book. Let μ be a finite positive Borel measure on the complex plane C with μ(C) = n. Let u(z) be the logarithmic potential of μ: u ( z )Characteristic function (probability theory) (5,208 words) [view diff] exact match in snippet view article
then the domain of the characteristic function can be extended to the complex plane, and φ X ( − i t ) = M X ( t ) . {\displaystyle \varphi _{X}(-it)=M_{X}(t)Dehn twist (749 words) [view diff] exact match in snippet view article find links to article
< 1 } {\displaystyle a(0;0,1)=\{z\in \mathbb {C} :0<|z|<1\}} in the complex plane. By extending to the torus the twisting map ( e i θ , t ) ↦ ( e i (Kalles Fraktaler (478 words) [view diff] exact match in snippet view article find links to article
for showing iteration depths or entering location parameters in the complex plane. The via zooming reached location can be saved as a KFR file. The renderedComposition operator (745 words) [view diff] exact match in snippet view article find links to article
operator. When the polynomials are orthogonal on some region of the complex plane (viz, in Bergman space), the Jacobi operator is replaced by a HessenbergInversive geometry (4,386 words) [view diff] exact match in snippet view article find links to article
the whole plane and so are necessarily conformal. Consider, in the complex plane, the circle of radius r {\displaystyle r} around the point a {\displaystyleUnivalent (91 words) [view diff] exact match in snippet view article find links to article
function – an injective holomorphic function on an open subset of the complex plane Univalent foundations – a type-based approach to foundation of mathematicsTrefoil knot (1,260 words) [view diff] exact match in snippet view article find links to article
obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic)Julia set (5,717 words) [view diff] exact match in snippet view article find links to article
} ) Then the filled Julia set for this system is the subset of the complex plane given by K ( f c ) = { z ∈ C : ∀ n ∈ N , | f c n ( z ) | ≤ R } , {\displaystyleFuchsian model (662 words) [view diff] exact match in snippet view article find links to article
either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of theResonance (7,906 words) [view diff] exact match in snippet view article find links to article
For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonateFredholm operator (1,472 words) [view diff] exact match in snippet view article find links to article
T ) {\displaystyle H^{2}(\mathbf {T} )} on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complexPolylogarithm (10,165 words) [view diff] exact match in snippet view article find links to article
notation without the subscript. Different polylogarithm functions in the complex plane Li –3(z) Li –2(z) Li –1(z) Li0(z) Li1(z) Li2(z) Li3(z) The polylogarithmGyrator–capacitor model (2,689 words) [view diff] exact match in snippet view article find links to article
P=r_{\mathrm {M} }I_{\mathrm {M} }^{2}} The magnetic effective resistance on a complex plane appears as the side of the resistance triangle for magnetic circuitLocal martingale (1,610 words) [view diff] exact match in snippet view article find links to article
| u − 1 | {\displaystyle u\mapsto \ln |u-1|} is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as τ k = minMontel's theorem (577 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \bigcup _{f\in {\mathcal {F}}}f(U)} is dense in the complex plane. The stronger version of Montel's theorem (occasionally referred toInterior (topology) (2,251 words) [view diff] exact match in snippet view article
\operatorname {int} \mathbb {Q} =\varnothing .} If X {\displaystyle X} is the complex plane C , {\displaystyle \mathbb {C} ,} then int ( { z ∈ C : | z | ≤ 1 }Wrapped Cauchy distribution (2,037 words) [view diff] exact match in snippet view article find links to article
. Viewing the z n {\displaystyle z_{n}} as a set of vectors in the complex plane, the R ¯ 2 {\displaystyle {\overline {R}}^{2}} statistic is the lengthSpherical wave transformation (7,711 words) [view diff] exact match in snippet view article find links to article
group of the plane (corresponding to the Möbius group of the extended complex plane) is isomorphic to the Lorentz group. A special case of Lie sphere geometryPrüfer group (1,137 words) [view diff] exact match in snippet view article find links to article
The Prüfer 2-group with presentation ⟨gn: gn+12 = gn, g12 = e⟩, illustrated as a subgroup of the unit circle in the complex planeBackward differentiation formula (1,077 words) [view diff] exact match in snippet view article find links to article
the plots below. Ideally, the region contains the left half of the complex plane, in which case the method is said to be A-stable. However, linear multistepAutomorphic form (1,652 words) [view diff] exact match in snippet view article find links to article
The Dedekind eta-function is an automorphic form in the complex plane.Principal ideal (1,470 words) [view diff] exact match in snippet view article find links to article
non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider ( a , b ) = ( 2 , 0 ) {\displaystyle (a,b)=(2,0)} and ( 1Z function (1,384 words) [view diff] exact match in snippet view article find links to article
Z function in the complex plane, plotted with a variant of domain coloring.Extended real number line (2,205 words) [view diff] exact match in snippet view article find links to article
} on the projectively extended real line. Division by zero Extended complex plane Extended natural numbers Improper integral Infinity Log semiring SeriesWeierstrass function (2,395 words) [view diff] exact match in snippet view article find links to article
February 2017. Retrieved 28 July 2006. Weierstrass function in the complex plane Archived 24 September 2009 at the Wayback Machine Beautiful fractalToeplitz operator (547 words) [view diff] exact match in snippet view article find links to article
Hardy space. Let S 1 {\displaystyle S^{1}} be the unit circle in the complex plane, with the standard Lebesgue measure, and L 2 ( S 1 ) {\displaystyleJordan normal form (6,847 words) [view diff] exact match in snippet view article find links to article
describe multiplication by λ i {\displaystyle \lambda _{i}} in the complex plane. The superdiagonal blocks are 2×2 identity matrices and hence in thisGábor Szegő (1,196 words) [view diff] exact match in snippet view article find links to article
(1936). "On some Hermitian forms associated with two given curves of the complex plane". Trans. Amer. Math. Soc. 40 (3): 450–461. doi:10.1090/s0002-9947-1936-1501884-1Bernhard Riemann (2,959 words) [view diff] exact match in snippet view article find links to article
Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection betweenReiss relation (289 words) [view diff] exact match in snippet view article find links to article
the points of a plane algebraic curve meeting a given line. If C is a complex plane curve given by the zeros of a polynomial f(x,y) of two variables, andFundamental parallelogram (50 words) [view diff] exact match in snippet view article find links to article
Fundamental parallelogram may mean: Fundamental pair of periods on the complex plane Primitive cell on the Euclidean plane This disambiguation page listsCurrent algebra (832 words) [view diff] exact match in snippet view article find links to article
precise mathematical definition in terms of integration contours on the complex plane, thus avoiding some of the formal divergence difficulties commonly encounteredFractal string (2,517 words) [view diff] exact match in snippet view article find links to article
geometric zeta function is not defined to all of the complex plane, we take a subset of the complex plane called the "window", and look for the "visible" complexVon Mises distribution (2,351 words) [view diff] exact match in snippet view article find links to article
. Viewing the z n {\displaystyle z_{n}} as a set of vectors in the complex plane, the R ¯ 2 {\displaystyle {\bar {R}}^{2}} statistic is the square ofOrthogonal polynomials (2,060 words) [view diff] exact match in snippet view article find links to article
One can also consider orthogonal polynomials for some curve in the complex plane. The most important case (other than real intervals) is when the curveAdolf Busemann (908 words) [view diff] exact match in snippet view article find links to article
reduced the complexity of the airflow to a conformal mapping in the complex plane, and was used for some time in the industry.[citation needed] BusemannAddition (9,554 words) [view diff] exact match in snippet view article find links to article
complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex planePolar set (potential theory) (296 words) [view diff] exact match in snippet view article
ISBN 0-88275-224-3. Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: CambridgeQuasiconformal mapping (1,324 words) [view diff] exact match in snippet view article find links to article
under composition. The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compactBandlimiting (1,080 words) [view diff] exact match in snippet view article find links to article
polynomial. All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that allLatitude (6,184 words) [view diff] exact match in snippet view article find links to article
however, involve a generalisation of the conformal latitude to the complex plane). The isometric latitude, ψ, is used in the development of the ellipsoidalWright omega function (551 words) [view diff] exact match in snippet view article find links to article
237147028\\\end{array}}} Plots of the Wright omega function on the complex plane z = Re(ω(x + i y)) z = Im(ω(x + i y)) ω(x + i y) Not to be confusedBandlimiting (1,080 words) [view diff] exact match in snippet view article find links to article
polynomial. All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that allHermitian symmetric space (7,418 words) [view diff] exact match in snippet view article find links to article
disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dualGegenbauer polynomials (1,339 words) [view diff] exact match in snippet view article find links to article
Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 functionStruve function (1,969 words) [view diff] exact match in snippet view article find links to article
Plot of the Struve function H n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3DElliptic curve (8,439 words) [view diff] exact match in snippet view article find links to article
affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) toBicomplex number (1,759 words) [view diff] exact match in snippet view article find links to article
numbers: the representation of two-dimensional potential flows in the complex plane and the complex exponential function. M.E. Luna-Elizarrarás, M. ShapiroAsymptotic analysis (2,774 words) [view diff] exact match in snippet view article find links to article
_{n=0}^{\infty }w^{n}} The expression on the left is valid on the entire complex plane w ≠ 1 {\displaystyle w\neq 1} , while the right hand side convergesGenerating set of a group (1,746 words) [view diff] exact match in snippet view article find links to article
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group.D (disambiguation) (1,132 words) [view diff] exact match in snippet view article
{\displaystyle \mathbb {D} } in blackboard bold, the unit disk in the complex plane, or the decimal fractions; see Number Cohen's d, a statistical measureScientific visualization (3,071 words) [view diff] exact match in snippet view article find links to article
interactions performed in 3D. In complex analysis, functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projectionEuler's rotation theorem (4,498 words) [view diff] exact match in snippet view article find links to article
that the eigenvalues of a unitary matrix are on the unit circle in the complex plane: α k ∗ = 1 α k ⟺ α k ∗ α k = | α k | 2 = 1 , k = 1 , … , m . {\displaystyleCovering space (6,980 words) [view diff] exact match in snippet view article find links to article
C {\displaystyle \mathbb {C} } the complex plane and C × {\displaystyle \mathbb {C} ^{\times }} the complex plane minus the origin. Then the map p : CSteven G. Krantz (1,915 words) [view diff] exact match in snippet view article find links to article
Conformal Geometry is a study of classical conformal geometry in the complex plane, and is the first Dover book that is not a reprint of a classic butDimension (3,932 words) [view diff] exact match in snippet view article find links to article
The complex plane can be mapped to the surface of a sphere, called the Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mappedAmerican Invitational Mathematics Examination (1,004 words) [view diff] exact match in snippet view article find links to article
{\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} in the complex plane are the vertices of a right triangle with hypotenuse h {\displaystyleComplex squaring map (521 words) [view diff] no match in snippet view article find links to article
and Applications, 10(11):949–953. Wikibooks has a book on the topic of: Fractals/Iterations_in_the_complex_plane/q-iterations#Dynamic_plane_for_c.3D00 (8,222 words) [view diff] exact match in snippet view article find links to article
are extended to form the complex numbers, 0 becomes the origin of the complex plane. The number 0 can be regarded as neither positive nor negative or, alternativelyPower rule (2,650 words) [view diff] exact match in snippet view article find links to article
complex number and z {\displaystyle z} is a complex number in a slit complex plane that excludes the branch point of 0 and any branch cut connected toSylvester's law of inertia (1,431 words) [view diff] exact match in snippet view article find links to article
same number of eigenvalues on each open ray from the origin in the complex plane. Metric signature Morse theory Cholesky decomposition Haynsworth inertiaGrace–Walsh–Szegő theorem (177 words) [view diff] exact match in snippet view article find links to article
affine in each variable separately. Let A be a circular region in the complex plane. If either A is convex or the degree of ƒ is n, then for every ζ 1Quadratic function (2,884 words) [view diff] exact match in snippet view article find links to article
about its vertex (orange). Its x-intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane (green).Integral domain (3,124 words) [view diff] exact match in snippet view article find links to article
integral domain. If U {\displaystyle U} is a connected open subset of the complex plane C {\displaystyle \mathbb {C} } , then the ring H ( U ) {\displaystyleArakawa–Kaneko zeta function (387 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \xi _{k}(s)} has analytic continuation to the whole complex plane as an entire function. The special case k = 1 gives ξ 1 ( s ) = s ζModular lambda function (3,503 words) [view diff] exact match in snippet view article find links to article
Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard inOrr–Sommerfeld equation (2,830 words) [view diff] exact match in snippet view article find links to article
the form λ = − i α c {\displaystyle \lambda =-i\alpha {c}} ) in the complex plane. The rightmost eigenvalue is the most unstable one. At the criticalDeming regression (1,527 words) [view diff] exact match in snippet view article find links to article
a point z j = x j + i y j {\displaystyle z_{j}=x_{j}+iy_{j}} in the complex plane (i.e., the point ( x j , y j ) {\displaystyle (x_{j},y_{j})} where iBulirsch–Stoer algorithm (627 words) [view diff] exact match in snippet view article find links to article
is rather far outside a circle around the known data points in the complex plane, rational function interpolation or extrapolation can have remarkable