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Location around which a function displays irregular behavior
essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is
Essential_singularity
Point where a mathematical object behaves irregularly
singularity is removable). The point a {\displaystyle a} is an essential singularity of f {\displaystyle f} if it is neither a removable singularity nor
Singularity_(mathematics)
Hypothetical event
The technological singularity, often simply called the singularity, is a hypothetical event in which technological growth accelerates beyond human control
Technological_singularity
Theorem about the range of an analytic function
Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle w} , then on any punctured neighborhood
Picard_theorem
Power series with negative powers
highest term; on the other hand, if f {\displaystyle f} has an essential singularity at c {\displaystyle c} , the principal part is an infinite sum (meaning
Laurent_series
Attribute of a mathematical function
\over z(z-1)}} it is apparent that the singularity at z = 0 {\displaystyle z=0} is a removable singularity and then the residue at z = 0 {\displaystyle
Residue_(complex_analysis)
Mathematical theorem
on U ∖ { z 0 } {\displaystyle U\setminus \{z_{0}\}} , but has an essential singularity at z 0 {\displaystyle z_{0}} . The Casorati–Weierstrass theorem
Casorati–Weierstrass_theorem
Concept of complex analysis
limit does not exist, then f {\displaystyle f} instead has an essential singularity at c {\displaystyle c} . If the limit is 0 {\displaystyle 0}
Residue_theorem
Has no other singularities close to it
function, then a {\displaystyle a} is an isolated singularity of f {\displaystyle f} . Every singularity of a meromorphic function on an open subset U
Isolated_singularity
Concept in complex analysis
certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function
Zeros_and_poles
Undefined point on a holomorphic function which can be made regular
{\text{sinc}}(z)={\frac {\sin z}{z}}} has a singularity at z = 0 {\displaystyle z=0} . This singularity can be removed by defining sinc ( 0 ) := 1
Removable_singularity
Class of mathematical function
singularity. The function f ( z ) = sin 1 z {\displaystyle f(z)=\sin {\frac {1}{z}}} is not meromorphic either, as it has an essential singularity at
Meromorphic_function
Point of interest for complex multi-valued functions
which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term branch
Branch_point
Mathematical theory
mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable
Singularity_theory
Type of function in mathematics
not a failure of convergence of the power series, nor a pole or essential singularity, but the branching of the analytic continuation. In effect, z =
Analytic_function
Theorem in complex analysis
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Cauchy's_integral_theorem
Branch of mathematics studying functions of a complex variable
functions near essential singularities is described by Picard's theorem. Functions that have only poles but no essential singularities are called meromorphic
Complex_analysis
Mathematical analysis of discontinuous points
or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables)
Classification of discontinuities
Classification_of_discontinuities
Mathematical theorem in complex analysis
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Maximum_modulus_principle
Seven mathematical problems with a US$1 million prize for each solution
S2CID 216323223. Theorem 2 implies that ζ {\displaystyle \zeta } has an essential singularity at infinity Bombieri, Enrico (2006). "The Riemann hypothesis" (PDF)
Millennium_Prize_Problems
Polish mathematician and physicist (1909–1984)
changes in human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know
Stanisław_Ulam
Analytic function in mathematics
complex infinity on the Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer
Riemann_zeta_function
Theorem about zeros of holomorphic functions
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Rouché's_theorem
Theorem in complex analysis
z-z_{Z}}+{g'(z) \over g(z)}.} Since g(zZ) ≠ 0, it follows that g' (z)/g(z) has no singularities at zZ, and thus is analytic at zZ, which implies that the residue of
Argument_principle
Number with a real and an imaginary part
of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0. Complex numbers have applications in
Complex_number
Right conoid ruled surface
{\displaystyle z={\frac {2xy}{x^{2}+y^{2}}}.} This function has an essential singularity at the origin. By using cylindrical coordinates in space, we can
Plücker's_conoid
In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable"
Movable_singularity
Widely-used term in mathematics
0 {\displaystyle 0} , then f ( z ) {\displaystyle f(z)} has an essential singularity at a {\displaystyle a} if and only if the principal part is an infinite
Principal_part
Theorem in complex analysis
{\displaystyle \mathbb {C} \cup \{\infty \}} . Viewed this way, the only possible singularity for entire functions, defined on C ⊂ C ∪ { ∞ } {\displaystyle \mathbb
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Mathematical function that preserves angles
often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the Big
Conformal_map
Functions in mathematics
harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution
Harmonic_function
Statement in complex analysis
lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry. Let D =
Schwarz_lemma
Complex-differentiable (mathematical) function
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Holomorphic_function
Number of times a curve wraps around a point in the plane
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Winding_number
Second-order partial differential equation
only valid locally, or provided that the path does not loop around a singularity. For example, if r and θ are polar coordinates and φ = log r , {\displaystyle
Laplace's_equation
analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati
List of complex analysis topics
List_of_complex_analysis_topics
Increase in the rate of technological change through history
century, leading to a singularity. Kurzweil elaborates on his views in his books The Age of Spiritual Machines and The Singularity Is Near. In the natural
Accelerating_change
Provides integral formulas for all derivatives of a holomorphic function
. This is analytic (since the contour does not contain the other singularity). We can simplify f 1 {\displaystyle f_{1}} to be: f 1 ( z ) = z 2 z
Cauchy's_integral_formula
Geometric representation of the complex numbers
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Complex_plane
Type of generalized function
f is any function that is holomorphic everywhere except for an essential singularity at 0 (for example, e1/z), then ( f , − f ) {\displaystyle (f,-f)}
Hyperfunction
One-dimensional complex manifold
puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity)
Riemann_surface
bang Stephen W. Hawking (1942–2018) described singularities in general relativity and developed singularity-free models of the big bang; predicted primordial
List_of_cosmologists
Theorem on holomorphic functions
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Open mapping theorem (complex analysis)
Open_mapping_theorem_(complex_analysis)
Hungarian and American mathematician and physicist (1903–1957)
189–191. The Technological Singularity by Murray Shanahan, (MIT Press, 2015), page 233 Chalmers, David (2010). "The singularity: a philosophical analysis"
John_von_Neumann
Characteristic property of holomorphic functions
hypothesis of real differentiability at the point z 0 {\displaystyle z_{0}} is essential and cannot be dispensed with. For example, the function f ( x , y ) =
Cauchy–Riemann_equations
Integral criterion for holomorphy
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Morera's_theorem
Components of the Fatou set
Baker domain: these are "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" one example
Classification of Fatou components
Classification_of_Fatou_components
Mathematical theorem
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Riemann_mapping_theorem
Theorem
center a {\displaystyle a} to the nearest non-removable singularity; if there are no singularities (i.e., if f {\displaystyle f} is an entire function),
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
French mathematician (1856–1941)
function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity. He made important
Émile_Picard
Mathematical functions which are smooth but not analytic
\{0\}\ni z\mapsto e^{-{\frac {1}{z}}}\in \mathbb {C} ,} has an essential singularity at the origin, and hence is not even continuous, much less analytic
Non-analytic_smooth_function
Function that is holomorphic on the whole complex plane
entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire
Entire_function
Topics referred to by the same term
Weierstrass–Casorati theorem describes the behavior of holomorphic functions near essential singularities The Weierstrass preparation theorem describes the behavior of analytic
Weierstrass_theorem
Concept in complex analysis
University Press. ISBN 0-521-28763-4. Alan D Solomon (Jan 1, 1994). The Essentials of Complex Variables I. Research & Education Assoc. ISBN 0-87891-661-X
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Division of mathematical analysis
grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for analytic functions (and meromorphic functions)
Value distribution theory of holomorphic functions
Value_distribution_theory_of_holomorphic_functions
Quantum lattice model
transition. The KT transition predicts that the free energy has an essential singularity that goes like e − c | g − g c | {\displaystyle e^{-{\tfrac {c}{\sqrt
Quantum_clock_model
Gender-neutral English pronoun
Singular they is a gender-neutral third-person pronoun in English. It typically occurs with an indeterminate antecedent, to refer to an unknown person
Singular_they
Used to count, measure, and label
distinguishing between poles and branch points, and introduced the concept of essential singular points.[clarification needed] This eventually led to the concept of
Number
Theorem in complex analysis
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Borel–Carathéodory_theorem
General relativity model near spacetime singularities
relativity has a page on the topic of: BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe
BKL_singularity
Expression which is not assigned an interpretation
function is undefined, is called a singularity. Some different types of singularities include: Removable singularities - in which the function can be extended
Undefined_(mathematics)
Conformal mappings in complex analysis
regular singular points at z = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points
Schwarz_triangle_function
Analytic function on the upper half-plane with a certain behavior under the modular group
its q-expansion. It can only have at most a pole at q = 0, not an essential singularity as exp(1/q) has. Here, a matrix ( a b c d ) {\displaystyle
Modular_form
Harmonic functions as solutions to Laplace's equation
Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions
Potential_theory
Infinite sum that is considered independently from any notion of convergence
formula Basic theory Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles Complex functions Complex-valued
Formal_power_series
discontinuity of the second kind. (This is distinct from the term essential singularity which is often used when studying functions of complex variables
Glossary_of_calculus
Sequence of differential equation solutions
origin once in a counterclockwise direction without enclosing the essential singularity at 1 The addition formula for Laguerre polynomials: L n ( α 1 +
Laguerre_polynomials
Hypothetical object of spacetime
general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy, matter, light
White_hole
Type of perturbation problem
because the function e − 1 / z {\displaystyle e^{-1/z}} possesses an essential singularity at z = 0 {\displaystyle z=0} in the complex z {\displaystyle z}
Perturbation problem beyond all orders
Perturbation_problem_beyond_all_orders
Italian mathematician (1835–1890)
essential singularities, which is that every holomorphic function gets values from any complex neighbourhood, in any neighbourhood of the singularity
Felice Casorati (mathematician)
Felice_Casorati_(mathematician)
from the original (PDF) on 2011-06-23. Retrieved February 12, 2013. Isaac M. Horowitz: An essential singularity in the complex domain of control theory
Isaac_Horowitz
Special functions in mathematics
differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on the projective line P 1 {\displaystyle
Painlevé_transcendents
for example exponential ) infinity is not a fixed point but an essential singularity and there is no Boettcher isomorphism. Here dynamic ray is defined
External_ray
distinguishes between poles and branch points and introduces the concept of essential singular points. 1850 – George Gabriel Stokes rediscovers and proves Stokes'
Timeline_of_mathematics
Innate constituent character-aspects within the soul, in Hasidism
consciousness. The quality of Faith reflects the Etzem-essential singular point of the soul, beyond the essential powers of Will and Delight. Above-conscious Delight
Kochos_hanefesh
Ideologies of change via capitalism and technology
self-revolutionizing capitalism that would culminate in a technological singularity, resulting in artificial intelligence surpassing and eliminating humanity
Accelerationism
Concept in complex analysis
doi:10.1007/BF02419336, JFM 41.0487.01, S2CID 122678686. "Studies on essential singular points of analytic functions of two or more complex variables" (English
Wirtinger_derivatives
Mathematical technique for improving convergence
{\displaystyle f(z)} can have singularities in the complex plane (branch point singularities, poles or essential singularities), which limit the radius of
Series_acceleration
Function with two complex number "periods"
Liouville's theorem. Since the function is meromorphic, it has no essential singularities and its poles are isolated. Therefore a translated lattice that
Doubly_periodic_function
Matrix used in complex analysis
another derivation of the Grunsky inequalities using reproducing kernels and singular integral operators in geometric function theory; a more recent related
Grunsky_matrix
distinguishes between poles and branch points and introduces the concept of essential singular points, 1850 - George Gabriel Stokes rediscovers and proves Stokes'
Timeline of calculus and mathematical analysis
Timeline_of_calculus_and_mathematical_analysis
Method of determining a point in 3D space
\mathbf {C} _{1},\mathbf {C} _{2}} . A point in this subset is then a singularity of the triangulation method. The reason for the failure can be that some
Triangulation (computer vision)
Triangulation_(computer_vision)
} is entire and injective. If ∞ {\displaystyle \infty } were an essential singularity of F {\displaystyle F} , Picard implies F {\displaystyle F} is dense
Fatou–Bieberbach_domain
systems of linear differential equations, all with the same (generic) singularity structure. One therefore allows the matrices A j ( i ) {\displaystyle
Isomonodromic_deformation
Mathematical technique in aerodynamics
1} the PG transformation features a singularity. The singularity is also called the Prandtl–Glauert singularity, and the flow resistance is calculated
Prandtl–Glauert transformation
Prandtl–Glauert_transformation
Country in South Asia
Governor General of the East India Company set the stage for changes essential to a modern state: the consolidation and demarcation of sovereignty, the
India
2021 book by Oliver Krüger
Virtual Immortality – God, Evolution, and the Singularity in Post- and Transhumanism is a study by German religious scholar Oliver Krüger. Krüger traces
Virtual Immortality – God, Evolution, and the Singularity in Post- and Transhumanism
Virtual_Immortality_–_God,_Evolution,_and_the_Singularity_in_Post-_and_Transhumanism
Differentiable manifold
variables. An English translation of the title reads as: "studies on essential singular points of analytic functions of two or more complex variables". Boggess
CR_manifold
Bearer of truth values
This raises the question of whether being affirmative or negative is an essential feature of propositions at the level of content rather than a linguistic
Proposition
Inputs for which a function's value is non-zero
are equal μ {\displaystyle \mu } -almost everywhere. In that case, the essential support of a measurable function f : X → R {\displaystyle f:X\to \mathbb
Support_(mathematics)
Concept in computer vision
In computer vision, the essential matrix is a 3 × 3 {\displaystyle 3\times 3} matrix, E {\displaystyle \mathbf {E} } that relates corresponding points
Essential_matrix
Linear operator equal to its own adjoint
{\displaystyle h} , then the spectrum of T {\displaystyle T} is just the essential range of h {\displaystyle h} . More complete versions of the spectral
Self-adjoint_operator
State of being real
Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does not know
Existence
2026 American film
a score of 8 out of 10 and wrote that it "is an essential doc that reveals the origins of her singular voice with exceeding warmth and vulnerability."
Paralyzed by Hope: The Maria Bamford Story
Paralyzed_by_Hope:_The_Maria_Bamford_Story
English archaic 2nd person singular pronoun
when indicating singularity to avoid confusion was needed; concurrently, the plural forms, ye and you, began to also be used for singular: typically for
Thou
1957 studio album by Miles Davis
react with ingenuity and precision is expressed in the consistency and singularity of each solo as it is maintained from one musician to the next without
Cookin' with the Miles Davis Quintet
Cookin'_with_the_Miles_Davis_Quintet
Aspect of mathematical spectrum theory
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of
Essential_spectrum
2026 Marvel Studios television special
finding some purpose. Bernthal also said the special would get to "the essential question of who he is". Deborah Ann Woll as Karen Page: A former reporter
The_Punisher:_One_Last_Kill
"Nothingness" in Kabbalah and Hasidic philosophy
called the "singularity". Hanson says that although Hebrew letters have shapes they are actually made out of nothing, as well as the singularity of the Big
Ayin_and_Yesh
distinguishes between poles and branch points and introduces the concept of essential singular points. J. J. Sylvester originates the term matrix in mathematics
1850_in_science
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
Boy/Male
Indian
Singularity
Female
English
English name derived from the vocabulary word, from Latin essentia, ESSENCE means "essence; being."
Boy/Male
Muslim
Singularity
Girl/Female
Arabic, Muslim
Imperative; Essential
Girl/Female
Arabic, Muslim, Sindhi
Singularity
Girl/Female
Muslim/Islamic
Singularity
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
Girl/Female
Indian
Henna
Boy/Male
Arabic
Flower in Heaven
Boy/Male
Hindu, Indian, Punjabi, Sikh
Absorbed in the Lord
Girl/Female
Hindu
Goddess of gold
Girl/Female
Tamil
Manvita | மாநவிதா, மாநவீதாÂ
Most respectable
Boy/Male
Tamil
Sreethesh | ஸà¯à®°à¯€à®¤à¯‡à®·
Lord Vishnu
Boy/Male
Afghan, Arabic, Danish, French, Kashmiri, Muslim, Pashtun, Sindhi
Singular; Unique; Alone; Exclusively; Unequalled; Exceptional; Peerless
Girl/Female
English
The first appearance of daylight; daybreak.
Boy/Male
Polish
Declares peace.
Girl/Female
Gujarati, Hindu, Indian
Cool; Cold
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
ESSENTIAL SINGULARITY
a.
Containing the essence or characteristic portion of a substance, as of a plant; highly rectified; pure; hence, unmixed; as, an essential oil.
a.
Not essential; unessential.
n. pl.
Essential parts.
n.
An essential element; a deciding point, fact, or consideration; an essential or influential circumstance.
a.
Not essential.
n.
The quality of being essential; the essential part.
a.
Necessary; indispensable; -- said of those tones which constitute a chord, in distinction from ornamental or passing tones.
v. t.
To deprive of anything essential.
n.
That which is essential; first or constituent principle; as, the essentials or religion.
adv.
In an essential manner or degree; in an indispensable degree; really; as, essentially different.
a.
Hence, really existing; existent.
n.
Existence; being.
a.
Not essential; not of prime importance; not indispensable; unimportant.
a.
Very necessary; highly important; essential.
n.
Essential quality; property; attribute.
a.
Belonging to the essence, or that which makes an object, or class of objects, what it is.
a.
Idiopathic; independent of other diseases.
n.
A thing not essential.
n.
Essential element, or constituent element.
a.
Important in the highest degree; indispensable to the attainment of an object; indispensably necessary.