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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 (4,144 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'sGamma 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 meromorphicError function (7,328 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 functionPoint 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 projectivePoint 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 projectiveAiry function (4,053 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 functionsBloch 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 ) | {\displaystyleHyperbolic functions (4,989 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 transcendentalUniform boundedness (400 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 isUnit 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 (2,002 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 holomorphicSubharmonic 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 (7,636 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 eachScorer'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 planeFunctional 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)Trigonometric 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 (3,429 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 contourNevanlinna theory (2,609 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), statesPrime omega function (4,100 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 factorsSchwarz 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 describedResolvent formalism (871 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 residueLine 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 homotopyEuler product (2,226 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, ifFatou 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. ŚwiątekPosition operator (2,611 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 \mathbbMonodromy (1,692 words) [view diff] exact match in snippet view article find links to article
{\displaystyle F(z)} in some open subset E {\displaystyle E} of the punctured complex plane C ∖ { 0 } {\displaystyle \mathbb {C} \backslash \{0\}} may be continuedPainlevé transcendents (3,520 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)Stokes 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 asymptoticInfinity (6,114 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 resultingSimply 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 \mathbbStokes 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 asymptoticWeierstrass–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 {\displaystyleMethod of steepest descent (5,062 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 directionPlanar Riemann surface (5,233 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 exactGoogle 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 (AEuclidean 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 geometricalLehmer–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 forBiholomorphism (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)Continued fraction (8,708 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 introductionPoincaré 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 dFixed-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 holomorphicDragon curve (1,577 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}}}Jacobi elliptic functions (13,097 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 repeatingFunctional-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, makingBrown 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 (basedArakelyan'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 subsetDirichlet eta function (3,708 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:Nikolai 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 boundaryGeneralized Riemann hypothesis (1,330 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 (1,458 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, eachRigidity (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 pointExponential 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 ofConformal welding (1,128 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 domainsSelberg'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 thisPositive-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 definitionElliptic 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 fundamentalHypergeometric function (6,920 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 regularKelvin 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 ComplexPlot3DMittag-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 distributionLine 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 circleLambert W function (12,429 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 steppingFourier (369 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 FourierClassification 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 ( degCircle 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 ) {\displaystyleIndex 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 numbersFree particle (1,929 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 ) /Bijection (2,509 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 (injectionSingular 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 theseAnalog 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 differenceAmplitwist (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 thatSectrix 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 ratesJ-invariant (4,738 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 iffZ-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 ROCDe 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\inSplitting 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 trivialMonodromy 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 {\displaystyleBrennan 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 openGenerator (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 asRotation matrix (15,472 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 conventionCartan'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 }Artin L-function (2,047 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-functionsWave packet (5,930 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=4Donald 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} isHesse configuration (1,360 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 setAnger 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 ComplexPlot3DComplex 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}} . ThenList 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 holomorphicBorel 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: CambridgeDouady–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 isFibered 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 orMellin transform (4,681 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. ConditionsFilled 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-ReportEI (500 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 theBirch and Swinnerton-Dyer conjecture (3,131 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. ThisKenneth 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 belowMultiple time dimensions (1,070 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 ofSmooth 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} {\mathcalH 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 itHypocycloid (1,350 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, summingEuler'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 ) {\displaystyleChandrasekhar'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 astrophysicistLindsey–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-degreeSchwarzian derivative (7,160 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 3Function field of an algebraic variety (664 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 sayInfinite compositions of analytic functions (4,849 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 + aSubmersion (mathematics) (1,776 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 {\displaystyleAndrew 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 curveDissipation 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 theFundamental 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.Nevanlinna–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 finitelyHadamard (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;Bicorn (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} .Discrete 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 spectrumWrapped 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 equationHyperbolic orthogonality (1,291 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}Normal matrix (1,657 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 spectrumRamification (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 theoryAlternating-direction implicit method (3,498 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 (Logarithmic integral function (1,103 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 ComplexPlot3DXaoS (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 mediaNumerical range (2,893 words) [view diff] exact match in snippet view article find links to article
of W ( A ) {\textstyle W(A)} , then we can translate and rotate the complex plane so that the point translates to the origin, and the region W ( A ) {\textstyleNumerical range (2,893 words) [view diff] exact match in snippet view article find links to article
of W ( A ) {\textstyle W(A)} , then we can translate and rotate the complex plane so that the point translates to the origin, and the region W ( A ) {\textstyleMathematical visualization (764 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 projectionInversive 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 {\displaystyleTrefoil knot (1,287 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)Singular 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 ofResonance (7,933 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 antiresonateCartan'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 )Fredholm operator (1,476 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 complexDehn 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 (Polylogarithm (10,139 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 polylogarithmJulia set (5,718 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 theLocal 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 = minGyrator–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 circuitUnivalent (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 mathematicsWrapped 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 lengthComposition operator (756 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 HessenbergMontel's theorem (589 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 toSpherical wave transformation (7,733 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 geometryJordan normal form (6,956 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 thisAutomorphic 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.Backward 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 multistepAddition (10,120 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 planeExtended 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 SeriesInterior (topology) (2,257 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 }Z 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.Bernhard Riemann (2,926 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 betweenPrincipal 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 ( 1Weierstrass function (2,430 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 ) {\displaystyleFractal string (2,518 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" complexElliptic curve (8,433 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) toAdolf Busemann (909 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] BusemannCurrent 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 encounteredHermitian 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 dualThe Book of Numbers (math book) (402 words) [view diff] exact match in snippet view article
sequence, transcendental numbers, the Metonic cycle, combinatorics, the complex plane, nimbers, and surreal numbers. The Basic Library List Committee of theBandlimiting (1,158 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 allPower rule (2,643 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 toPrüfer 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 planeAsymptotic analysis (2,803 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 convergesVon Mises distribution (2,641 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 ofReiss 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 listsQuadratic function (2,893 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).Struve function (1,994 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 ComplexPlot3DD (disambiguation) (1,130 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 measurePolar 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: CambridgeGenerating 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.Euler's rotation theorem (4,497 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 (6,981 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 : CLatitude (6,166 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 ellipsoidalSteven G. Krantz (1,948 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 butAmerican Invitational Mathematics Examination (971 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 {\displaystyleGábor Szegő (1,201 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/s0002-9947-1936-1501884-1Dimension (3,930 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 mappedSylvester'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 inertiaIntegral domain (3,126 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 ) {\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.3D0Modular 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 in0 (8,232 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, alternativelyBicomplex number (1,806 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. ShapiroGrace–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 1Wright omega function (589 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 = ℜ { ω ( x + i y ) } {\displaystyle z=\Re \{\omega (x+iy)\}} z =Low-pass filter (4,696 words) [view diff] exact match in snippet view article find links to article
considering the pattern of poles and zeros of the Laplace transform in the complex plane. (In discrete time, one can similarly consider the Z-transform of the