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Concept in complex analysis
point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z0
Zeros_and_poles
Fine dining restaurant in Seoul, South Korea
Zero Complex (Korean: 제로 컴플렉스), stylized as ZERO COMPLEX, is a fine dining restaurant in Seoul, South Korea. Chef-owner Lee Choong Hu (이충후) studied at
Zero_Complex
Number property of being positive or negative
objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often
Sign_(mathematics)
Number with a real and an imaginary part
regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. It is common
Complex_number
Logarithm to the base of the mathematical constant e
negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more. The natural logarithm
Natural_logarithm
Angle of complex number about real axis
to write complex numbers in modulus-argument form. Hence for any complex number z, z=|z|eiArgz. This is only really valid if z is non-zero, but can be
Argument_(complex_analysis)
Representation of a quantum mechanical system
complex Hilbert space H. A pure state of a quantum system is represented by a non-zero vector ψ in H. The vectors ψ and λψ (with λ a non-zero complex
Bloch_sphere
One of Fredholm's theorems in mathematics
theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. If V {\displaystyle
Fredholm_alternative
Class of mathematical expression
In mathematics, division where the divisor (denominator) is zero is a problematic special case. Using fraction notation, the general example can be written
Division_by_zero
Generalized Euclidean space in mathematics
\mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes [ v ] {\displaystyle [v]} of non-zero vectors v ∈ H {\displaystyle
Projective_Hilbert_space
Whole number
numbers, real numbers, complex numbers, and algebraic structures. Multiplying any number by 0 results in 0; therefore, division by zero is generally regarded
0
2026 Japanese film by Takashi Yamazaki
Godzilla Minus Zero (Japanese: ゴジラ-0.0（マイナスゼロ）, Hepburn: Gojira Mainasu Zero) is a 2026 Japanese epic kaiju film written, directed, and with visual effects
Godzilla_Minus_Zero
Differentiating positive and negative zero
Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, and −0, +0 and 0 are three ways of writing the
Signed_zero
Removal of points from a manifold
projective plane with a single puncture). The set of non-zero complex numbers is the complex plane once punctured at the origin, sometimes denoted ℂ∗
Puncture_(topology)
Point where function's value is zero
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such
Zero_of_a_function
Used to count, measure, and label
ℕ⊂ℤ⊂ℚ⊂ℝ⊂ℂ⊂ℍ⊂𝕆⊂𝕊⊂𝕋⊂⋯ Alternatively, starting from the real numbers ℝ, which have zero complex units, this can be expressed as 𝒞0⊂𝒞1⊂𝒞2⊂𝒞3⊂𝒞4⊂𝒞5⊂⋯⊂𝒞n with 𝒞n
Number
Product of a number by itself
number is always a nonnegative real number, that is zero if and only if the complex number is zero. It is easier to compute than the absolute value (no
Square_(algebra)
Matrix equal to its conjugate-transpose
and the above follows, as a number can equal its complex conjugate only if their imaginary parts are zero. Only the main diagonal entries are necessarily
Hermitian_matrix
Special objects used in (mathematical) category theory
co-cones from F. In the category ChR of chain complexes over a commutative ring R, the zero complex is a zero object. In a short exact sequence of the form
Initial_and_terminal_objects
Property of a mathematical matrix
complex matrices, the most common definition says that M is positive-definite if and only if 𝐳∗M𝐳 is real and positive for every non-zero complex column
Definite_matrix
Number whose square is a given number
indeed a square root of −x, since(ix)2=i2(x)2=(−1)x=−x. For every non-zero complex number z there exist precisely two numbers w such that w2 = z: the principal
Square_root
Length in a vector space
real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains
Norm_(mathematics)
Arithmetic operation, inverse of nth power
treated as a complex number it has two imaginary square roots, +ix and −ix, where i is the imaginary unit. In general, any non-zero complex number has
Nth_root
Integer
square is −1 is −i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. In the
−1
Mathematical expression with disputed status
Zero to the power of zero, denoted as 00, is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics
Zero_to_the_power_of_zero
Topics referred to by the same term
Look up zero, 0, naught, nought, or 零 in Wiktionary, the free dictionary. Zero is a number. Zero, Zeros, or Zeroes may also refer to: Zero (Marvel Comics)
Zero_(disambiguation)
Complex-differentiable (mathematical) function
z¯, the complex conjugate of z, is zero: ∂f∂z¯=0, which is to say that, roughly, f is functionally independent from z¯, the complex conjugate
Holomorphic_function
Type of function in mathematics
integral powers, away from zero for negative integral powers, and for arbitrary complex powers on any open subset of the complex plane where the logarithm
Analytic_function
Situation where total gains match total losses
In game theory and economic theory, zero-sum game is a mathematical representation of a situation that involves two competing entities, where the result
Zero-sum_game
Systems security model
VPN connection is commonly not sufficient in the complex environment of a corporate network. The zero trust approach advocates mutual authentication, including
Zero_trust_architecture
Fundamental operation on complex numbers
since the real part of z⋅r‾ is zero only when the cosine of the angle between z and r is zero. Similarly, for a fixed complex unit u=eib, the equation z−z0z‾−z0‾=u2
Complex_conjugate
Theorem in complex analysis
function's zeros and poles, and an associated non-zero holomorphic function.[citation needed] It is clear that any finite set {cn} of points in the complex plane
Weierstrass factorization theorem
Weierstrass_factorization_theorem
Analytic function on the upper half-plane with a certain behavior under the modular group
a variable z, then F(Λ) is an analytic function of z. If α is a non-zero complex number and αΛ is the lattice obtained by multiplying each element of
Modular_form
generated by 1 and all logarithms of non-zero algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form β0+∑i=1nβilogαi
Four_exponentials_conjecture
Mathematical function, denoted exp(x) or e^x
mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Expression which is not assigned an interpretation
consistent set of mathematics referred to as the complex number plane. Therefore, within the discourse of complex numbers, −1 is in fact defined. Many new fields
Undefined_(mathematics)
Theory of a class of elliptic curves
where Z[i] is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It
Complex_multiplication
Conjecture on zeros of the zeta function
conjecture that the Riemann zeta function has its zeros only at the negative even integers and at complex numbers with real part 12. Many consider it to
Riemann_hypothesis
Diagram showing the singularities of a given control system's transfer function
processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain
Pole–zero_plot
Japanese light novel series and its adaptations
Re:Zero − Starting Life in Another World (Japanese: Re:ゼロから始める異世界生活, Hepburn: Ri:Zero kara Hajimeru Isekai Seikatsu), often referred to simply as Re:Zero
Re:Zero
Distance from zero to a number
from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Since the complex numbers
Absolute_value
Mathematical entity to describe the probability of each possible measurement on a system
itself). If a vector in a complex Hilbert space H can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in
Quantum_state
2026 video game
Zero Parades: For Dead Spies is a 2026 role-playing video game by ZA/UM. It is the studio's second game following Disco Elysium. It was released in May
Zero_Parades:_For_Dead_Spies
Analytic function in mathematics
definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution
Riemann_zeta_function
Algebraic operation on coordinate vectors
non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is
Dot_product
2017 video game
Horizon Zero Dawn is a 2017 action role-playing game developed by Guerrilla Games and published by Sony Interactive Entertainment. The first instalment
Horizon_Zero_Dawn
Topics referred to by the same term
type Convergence (goth festival), a goth festival C*, the group of non-zero complex numbers This disambiguation page lists articles associated with the title
C*_(disambiguation)
Principal square root of minus 1
every nonzero complex number has two distinct complex-valued square roots, which are additive inverses of each other, while zero has only zero as its (double)
Imaginary_unit
Method for producing composition algebras
the complex numbers a normed vector space over the real numbers: the norm of a complex number z is |z|=(z∗z)12. Furthermore, for any non-zero complex number z
Cayley–Dickson_construction
Every polynomial has a real or complex root
follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Function returning minus 1, zero or plus 1
given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as sgnx
Sign_function
Matrix that commutes with its conjugate transpose
to purely imaginary numbers. Invertible matrices are analogous to non-zero complex numbers. The inverse of a matrix has each eigenvalue inverted. A uniform
Normal_matrix
Mathematical expression
approaches zero quite rapidly. A linear fractional transformation (LFT) is a complex function of the form w=f(z)=az+bcz+d, where z is a complex variable
Continued_fraction
Generalization of a complex manifold that allows the use of singularities
Complex analytic varieties are analogous to algebraic varieties. Roughly speaking, a complex analytic variety is a zero locus of a set of a complex analytic
Complex_analytic_variety
Fine dining restaurant in Busan, South Korea
Korea afterwards. Before opening Palate, he worked as a sous chef at Zero Complex, a Michelin-starred restaurant in Seoul. The restaurant reportedly opened
Palate_(restaurant)
Class of mathematical function
In complex analysis, a meromorphic function on an open subset D {\displaystyle D} of the complex plane is a function that is holomorphic on all of D {\displaystyle
Meromorphic_function
Number which when multiplied by x equals 1
are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part
Multiplicative_inverse
Topics referred to by the same term
typesetting of some complex scripts Zero-width joiner Zero-width non-joiner Zero-width space Zero-width non-breaking space Zero waste, an environmental
0W
Non-printing character used in computerized typesetting
boxes, misplaced vowels or missing conjuncts instead of Indic text. The zero-width joiner (ZWJ, /ˈzwɪdʒ/; rendered: ; HTML entity: ‍ or ‍)
Zero-width_joiner
Theorem in complex analysis
Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an
Cauchy's_integral_theorem
Limit of roots of sequence of functions
mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally
Hurwitz's theorem (complex analysis)
Hurwitz's_theorem_(complex_analysis)
Branch of mathematics studying functions of a complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued
Complex_analysis
Type of mathematical expression
∑k=0nakxk That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of
Polynomial
Condition on transcendence of numbers
y2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then
Six_exponentials_theorem
Potential counterexample to the generalized Riemann hypothesis
of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as an exceptional zero, named after Edmund Landau and Carl Ludwig Siegel
Siegel_zero
Measure of temperature relative to absolute zero
is a physical quantity that measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic
Thermodynamic_temperature
a2\cdot b2=e2} Clifford gives a second example involving complex numbers. Given two non-zero complex numbers a and b, the following operation forms a right
Right_group
Algebraic curve in mathematics
curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number c to a lattice cΛ, then the corresponding curves are isomorphic
Elliptic_curve
Rate of human-caused greenhouse gas emissions
Global net-zero emissions are reached when greenhouse gas emissions and removals due to human activities are in balance. Net-zero emissions is often shortened
Net-zero_emissions
Statistical distribution of complex random variables
of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean:
Complex_normal_distribution
Mathematical function between groups that preserves multiplication structure
yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is
Group_homomorphism
Power series with negative powers
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative
Laurent_series
Affine space over the complex numbers
one-dimensional linear subspace (that is, a complex line through the origin of X). But if c is some non-zero complex number, then the set A of solutions of
Complex_affine_space
Theorem about zeros of holomorphic functions
|g(z)| < |f(z)| on ∂K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem
Rouché's_theorem
Japanese anime television series
Re:Zero − Starting Life in Another World (Japanese: Re:ゼロから始める異世界生活, Hepburn: Ri:Zero kara Hajimeru Isekai Seikatsu), often referred to simply as Re:Zero
Re:Zero_(TV_series)
Concept in class field theory
non-zero complex numbers, and for R {\displaystyle \mathbb {R} } it is a non-split extension of the Galois group of order 2 by the group of non-zero complex
Weil_group
Arithmetic operation
If the complex number is moved around zero by increasing its argument, after an increment of 2π, the complex number comes back to its initial position
Exponentiation
Proving validity without revealing other data
In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the
Zero-knowledge_proof
2026 release of AI-generated mathematical solutions
Siegel zeros, leading to several results in analytic number theory. A direct proof that Siegel zeroes do not exist, independent of the new zero-free half-planes
OpenAI bulk release of mathematical papers
OpenAI_bulk_release_of_mathematical_papers
Model of the extended complex plane plus a point at infinity
very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way
Riemann_sphere
Extension of the factorial function
to a meromorphic function defined for all complex numbers z, except integers less than or equal to zero. It is this extended version that is commonly
Gamma_function
Concept of complex analysis
singularities at any point in the complex plane), this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 = (z + i)(z − i)
Residue_theorem
Complex exponential in terms of sine and cosine
mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function
Euler's_formula
Geometric representation of the complex numbers
the positions of the zeros and poles of a function in the complex plane. The lengths of straight lines and curves in the complex plane represent real
Complex_plane
Function that is holomorphic on the whole complex plane
real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can
Entire_function
Video game series
Zero Escape, formerly released in Japan as Kyokugen Dasshutsu (Japanese: 極限脱出; lit. "Extreme Escape"), is a series of adventure games directed and written
Zero_Escape
Construction in homological algebra
complexes are derived intersections of zero loci. First, the Koszul complex Ks of (A,s) is a chain complex: the composition of any two maps is zero.
Koszul_complex
Theorem in complex analysis
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles
Argument_principle
Concept in algebraic geometry
provided by the multiplicative group C* of non-zero complex numbers that acts on an n-dimensional complex vector space Cn by scalar multiplication. In this
Geometric_invariant_theory
series Meromorphic function Entire function Pole (complex analysis) Zero (complex analysis) Residue (complex analysis) Isolated singularity Removable singularity
List of complex analysis topics
List_of_complex_analysis_topics
Set of coordinates where the coordinate hypersurfaces all meet at right angles
holomorphic function w = f(z) with non-zero complex derivative will produce a conformal mapping; if the resulting complex number is written w = u + iv, then
Orthogonal_coordinates
Definite integral of a scalar or vector field along a path
as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector
Line_integral
1990 racing game
1991, and in Europe in 1992. F-Zero is the first game in the F-Zero series and was a launch game for the SNES. F-Zero has been rereleased for the Virtual
F-Zero_(video_game)
2024 video game
Dragon Ball: Sparking! Zero is a 2024 fighting game developed by Spike Chunsoft and published by Bandai Namco Entertainment. Based on the Dragon Ball franchise
Dragon_Ball:_Sparking!_Zero
Matrix decomposition
±1 (or by any unit complex number). The compact SVD eliminates the rows and columns from 𝜮 which consist entirely of zeros, and also the superfluous
Singular_value_decomposition
Infinite sum that is considered independently from any notion of convergence
(This is different from the classical Laurent series of complex analysis.) For a non-zero formal Laurent series, the minimal integer n such that an≠0
Formal_power_series
Absence of the sensation of weight
sensation of weight, i.e., zero apparent weight. It is also termed zero g-force, or zero-g (named after the g-force) or, misleadingly, zero gravity. Weight is
Weightlessness
Number of times a curve wraps around a point in the plane
algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such
Winding_number
Group that is also a differentiable manifold with group operations that are smooth
numbers with multiplication), for complex numbers (because ℂ is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for
Lie_group
Exploring properties of the integers with complex analysis
methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the
Analytic_number_theory
travel, tourism, insurance
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