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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 cases

independent quantity. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by

unicritical polynomial, i.e. it has one finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity and

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's

function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple

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 projective

result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental

\mathbb {C} \cup \{\infty \}} be the one-point compactification of the complex plane C {\displaystyle \mathbb {C} } . In place of holomorphic functions defined

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 is

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 ) | {\displaystyle

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 parametrization

... As explained below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic behaviour of the Airy functions

mathematics, in the theory of ordinary differential equations in the complex plane C {\displaystyle \mathbb {C} } , the points of C {\displaystyle \mathbb

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 holomorphic

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 plane

|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 each

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 the

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 contour

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 taken

the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described

functions of one complex variable defined in a disc |z| ≤ R or in the whole complex plane (R = ∞). Subsequent generalizations extended Nevanlinna theory to algebroid

negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy

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)

In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors

position-line into the complex plane. It is nothing more than the canonical embedding of the real line into the complex plane. Note 2. The expected value

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 residue

\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 resulting

certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles)

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 continued

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, if

{L_{a}-\mu }{{\sqrt {2}}\sigma }}\right).\end{aligned}}} Plots in the complex plane The property erf (−z) = −erf z means that the error function is an odd

continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is C ∖ { 0 } , {\displaystyle \mathbb

approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction

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. Świątek

{\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 {\displaystyle

eigenvalues λ i {\displaystyle \lambda _{i}} of Google matrices in the complex plane at α = 1 {\displaystyle \alpha =1} for dictionary networks: Roget (A

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. Świątek

asymptotic behavior of functions can differ in different regions of the complex plane. This seemingly gives rise to a paradox when looking at the asymptotic

models in statistical mechanics. Given a parameter κ and a domain in the complex plane U, it gives a family of random curves in U, with κ controlling how much

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 exact

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical

enclosing roots like in the one-dimensional bisection method to the complex plane. It uses the Schur-Cohn test to test increasingly smaller disks for

continued fractions that are rapidly convergent almost everywhere in the complex plane. The long continued fraction expression displayed in the introduction

{\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)

of values that f ( z ) {\textstyle f(z)} assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original

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:

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 holomorphic

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 d

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}}}

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, the

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, making

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 subset

Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals

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, each

(only when χ {\displaystyle \chi } is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet

where increasing time has been mapped to some increasing radius on the complex plane. One is also interested in normal ordering of creation operators. A

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 (based

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 point

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 domains

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 of

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 definition

the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular

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 this

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 ComplexPlot3D

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 quadratrix

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 set

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 cases

}(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 {\displaystyle

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 repeating

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 repeating

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 boundary

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 circle

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 distribution

Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane. An injective non-surjective function (injection

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention

\Lambda } acting on C {\displaystyle \mathbb {C} } . Geometrically the complex plane is tiled with parallelograms. Everything that happens in one fundamental

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 ( deg

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 ) /

Unsolved problem in mathematics: In the complex plane, is it possible to "walk to infinity" in the Gaussian integers using the Gaussian primes as stepping

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, where

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 numbers

L} . For example, consider the analytification X {\displaystyle X} a complex plane curve Proj ( C [ x , y , z ] x n + y n + z n ) {\displaystyle

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 these

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 difference

{\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 trivial

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 rates

amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number z {\displaystyle z} such that

series, a special case of the Z-transform around the unit circle in the complex plane Discrete Fourier transform (DFT), occasionally called the finite Fourier

{\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\in

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 as

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 whole

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 {\displaystyle

ISBN 0-09-113411-0. Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: Cambridge

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. Conditions

a non-constant holomorphic function on a connected open set in the complex plane is an open mapping Open mapping theorem (topological groups), states

{\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 open

a non-trivial irreducible representation ρ is analytic in the whole complex plane. This is known for one-dimensional representations, the L-functions

circle. The region of convergence (ROC) is the set of points in the complex plane for which the Z-transform summation converges (i.e. doesn't blow up

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 ComplexPlot3D

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 }

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 iff

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 is

complex algebraic geometry. For instance, each singular point of a complex plane curve can be described topologically as the cone on a fibered knot or

ISBN 978-0-387-15851-8. Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report

functions are defined not only for the real values of argument(s), but to complex plane, and differential and/or integral representation, as well as expansions

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 3

the maxima of holomorphic functions within concentric circles in the complex plane. Hadamard three-lines theorem, concerning the maxima of holomorphic

horse species Exponential integral, a special function defined on the complex plane given the symbol Ei Education Index, a United Nations measure of the

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. This

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.

imaginary time dimension, changing time from a real number line into a complex plane. Introducing it into Minkowski spacetime allows a generalization of

{\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}

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 ) {\displaystyle

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 it

Theorem. A smooth 4-dimensional plane is isomorphic to the classical complex plane, or dim ⁡ Aut ⁡ P ≤ 6 {\displaystyle \dim \operatorname {Aut} {\mathcal

{\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 {\displaystyle

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 rendered

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 say

When representing the electrical circuit parameters as vectors in a complex plane, known as phasors, a capacitor's dissipation factor is equal to the

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 it

_{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 + a

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, summing

In atmospheric radiation, Chandrasekhar's H-function appears as the solutions of problems involving scattering, introduced by the Indian American astrophysicist

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-degree

A lattice in the complex plane and its fundamental domain, with quotient a torus.

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 finitely

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:

\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 spectrum

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} .

refer to: Entire function, a function that is holomorphic on the whole complex plane Entire (animal), an indication that an animal is not neutered Entire

Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation

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 below

complex plane; Hadamard three-circle theorem, a bound on the maximum modulus of complex analytic functions defined on an annulus in the complex plane;

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)

{\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 (

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} is

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 antiresonate

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 spectrum

operator. When the polynomials are orthogonal on some region of the complex plane (viz, in Bergman space), the Jacobi operator is replaced by a Hessenberg

⁡ | 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 = min

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 theory

the whole plane and so are necessarily conformal. Consider, in the complex plane, the circle of radius r {\displaystyle r} around the point a {\displaystyle

Plot of polygamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1's function ComplexPlot3D

. 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 length

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)

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 of

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 ComplexPlot3D

} on the projectively extended real line. Division by zero Extended complex plane Extended natural numbers Improper integral Infinity Log semiring Series

< 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 (

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 polylogarithm

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 circuit

}}} which can be presented as the one-point compactification of the complex plane C ∪ { ∞ } {\displaystyle \mathbb {C} \cup \{\infty \}} or as the complex

} ) 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 }   , {\displaystyle

\operatorname {int} \mathbb {Q} =\varnothing } If X {\displaystyle X} is the complex plane C , {\displaystyle \mathbb {C} ,} then int ⁡ ( { z ∈ C : | z | ≤ 1 }

several variables. In 2000 Browder published his article "Topology in the Complex Plane", which described the Brouwer fixed point theorem, the Jordan curve

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 )

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 the

Z function in the complex plane, plotted with a variant of domain coloring.

Technology 2003. Retrieved 28 July 2006. Weierstrass function in the complex plane Beautiful fractal. SpringerLink - Journal of Fourier Analysis and Applications

complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane

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 geometry

describe multiplication by λ i {\displaystyle \lambda _{i}} in the complex plane. The superdiagonal blocks are 2×2 identity matrices and hence in this

{\displaystyle \bigcup _{f\in {\mathcal {F}}}f(U)} is dense in the complex plane. The stronger version of Montel's Theorem (occasionally referred to

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 complex

function – an injective holomorphic function on an open subset of the complex plane Univalent foundations – a type-based approach to foundation of mathematics

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 multistep

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 ( 1

however, involve a generalisation of the conformal latitude to the complex plane). The isometric latitude, ψ, is used in the development of the ellipsoidal

One can also consider orthogonal polynomials for some curve in the complex plane. The most important case (other than real intervals) is when the curve

. 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 of

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" complex

angles, dimensions and proportions In complex analysis, functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projection

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, alternatively

complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (related to the Cauchy–Kovalevskaya theorem)

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 media

{\displaystyle \mathbb {D} } in blackboard bold, the unit disk in the complex plane, or the decimal fractions; see Number Cohen's d, a statistical measure

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) to

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 dual

under composition. The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact

_{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 converges

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 . {\displaystyle

interactions performed in 3D. In complex analysis, functions of the complex plane are inherently 4-dimensional, but there is no natural geometric projection

polynomial. All trigonometric polynomials are holomorphic on a whole complex plane, and there is a simple theorem in complex analysis that says that all

The Prüfer 2-group with presentation ⟨gn: gn+12 = gn, g12 = e⟩, illustrated as a subgroup of the unit circle in the complex plane

The Dedekind eta-function is an automorphic form in the complex plane.

The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group.

numbers: the representation of two-dimensional potential flows in the complex plane and the complex exponential function. M.E. Luna-Elizarrarás, M. Shapiro

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 function

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 ComplexPlot3D

C {\displaystyle \mathbb {C} } the complex plane and C × {\displaystyle \mathbb {C} ^{\times }} the complex plane minus the origin. Then the map p : C

The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group.

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 critical

the integral being a complex contour integral. Proof We regard the complex plane as R 2 {\displaystyle \mathbb {R} ^{2}} . Now, define u , v : R ¯ →

ISBN 0-88275-224-3. Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: Cambridge

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 confused

Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general. z w = z w {\displaystyle

integral domain. If U {\displaystyle U} is a connected open subset of the complex plane C {\displaystyle \mathbb {C} } , then the ring H ( U ) {\displaystyle

Fundamental parallelogram may mean: Fundamental pair of periods on the complex plane Primitive cell on the Euclidean plane This disambiguation page lists

same number of eigenvalues on each open ray from the origin in the complex plane. Metric signature Morse theory Cholesky decomposition Haynsworth inertia

Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between

precise mathematical definition in terms of integration contours on the complex plane, thus avoiding some of the formal divergence difficulties commonly encountered

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

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 in

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 to

{\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} in the complex plane are the vertices of a right triangle with hypotenuse h {\displaystyle

Lehmer–Schur algorithm, generalization of the bisection method in the complex plane Nested intervals Burden & Faires 1985, p. 31 "Interval Halving (Bisection)"

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 i

The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i