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Theorem about the range of an analytic function
Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. Little
Picard_theorem
Existence and uniqueness of solutions to initial value problems
known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst
Picard–Lindelöf_theorem
French mathematician (1856–1941)
Picard's little theorem states that every nonconstant entire function takes every value in the complex plane, with perhaps one exception. Picard's great
Émile_Picard
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
that the rank of NS(V) is finite is Francesco Severi's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V)
Picard_group
Symmetric holomorphic function
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 1879
Modular_lambda_function
Theorem in complex analysis
by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant. Liouville's theorem: Every
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Theorem in complex analysis
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard
Cauchy's_integral_theorem
Two theorems about families of holomorphic functions
version of Montel's theorem stated above is the analog of Liouville's theorem, while the second version corresponds to Picard's theorem. Montel space Fundamental
Montel's_theorem
Concept of complex analysis
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions
Residue_theorem
Theorem about metric spaces
can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first
Banach_fixed-point_theorem
Branch of mathematics studying functions of a complex variable
conditions (see Looman–Menchoff theorem). Holomorphic functions exhibit some remarkable features. For instance, Picard's theorem asserts that the range of an
Complex_analysis
Theorem about zeros of holomorphic functions
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K {\displaystyle
Rouché's_theorem
Mathematical function that preserves angles
complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types. The notion of conformality
Conformal_map
Phragmén–Lindelöf theorem (complex analysis) Picard theorem (complex analysis) Residue theorem (complex analysis) Riemann mapping theorem (complex analysis)
List_of_theorems
One-dimensional complex manifold
maps between Riemann surfaces, as detailed in Liouville's theorem and the Little Picard theorem: maps from hyperbolic to parabolic to elliptic are easy
Riemann_surface
Provides integral formulas for all derivatives of a holomorphic function
{f(z)}{z-a}}\,dz.} The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires f {\displaystyle f} to be complex differentiable
Cauchy's_integral_formula
Type of function in mathematics
of analytic functions are analytic is an easy consequence of Morera's theorem. The set A ∞ ( Ω ) {\displaystyle A_{\infty }(\Omega )} of all bounded
Analytic_function
Theorem regarding the existence of a solution to a differential equation
Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees
Peano_existence_theorem
Concept in complex analysis
Riemann–Roch theorem. Argument principle Control theory § Stability Filter design Filter (signal processing) Gauss–Lucas theorem Hurwitz's theorem (complex
Zeros_and_poles
Area of mathematics
considered as a generalization of Picard's theorem. Many other Picard-type theorems can be derived from the Second Fundamental Theorem. As another corollary from
Nevanlinna_theory
mathematical complex analysis, Schottky's theorem, introduced by Schottky (1904) is a quantitative version of Picard's theorem. It states that for a holomorphic
Schottky's_theorem
Integral criterion for holomorphy
mathematics, Morera's theorem, named after Giacinto Morera, gives a criterion for proving that a function is holomorphic. Morera's theorem states that a continuous
Morera's_theorem
Mathematical theorem
In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number
Riemann_mapping_theorem
Power series with negative powers
{\displaystyle \gamma } is an immediate consequence of Cauchy's integral theorem. One may also obtain the Laurent series for a complex function f ( z )
Laurent_series
Number of times a curve wraps around a point in the plane
the winding number in the complex plane are given by the following theorem: Theorem. Let γ : [ α , β ] → C {\displaystyle \gamma :[\alpha ,\beta ]\to \mathbb
Winding_number
Theorem in mathematics
forth. The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction
Inverse_function_theorem
Statement in complex analysis
z_{1}} . The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De Branges' theorem, formerly known as the Bieberbach
Schwarz_lemma
Mathematical theory
main theorems implies Picard's theorem, and the Second main theorem of Nevanlinna theory. Many other important generalizations of Picard's theorem can
Ahlfors_theory
Injective polynomial functions are bijective
implies surjectivity of f {\displaystyle f} . This is a corollary of Picard's theorem. The inverse of f {\displaystyle f} is always also a polynomial function
Ax–Grothendieck_theorem
Second-order partial differential equation
{\displaystyle u} is harmonic in D {\displaystyle D} , then the divergence theorem implies the compatibility condition ∫ ∂ D ∂ u ∂ ν d S = 0. {\displaystyle
Laplace's_equation
Study of differential field extensions induced by linear differential equations
necessary concepts and proved a rigorous version of this theorem. Kolchin (1952) extended Picard–Vessiot theory to partial differential fields (with several
Picard–Vessiot_theory
Theorem in topology
The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving
Brouwer_fixed-point_theorem
Characteristic property of holomorphic functions
{\partial (-v)}{\partial y}}=0.} Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily a conservative one on any simply-connected
Cauchy–Riemann_equations
17th-century conjecture proved by Andrew Wiles in 1994
Last Theorem within twenty-four hours. In the 1989 Star Trek: The Next Generation episode "The Royale", Captain Picard states that the theorem is still
Fermat's_Last_Theorem
according to this principle, Picard's theorem corresponds to Schottky's theorem, and Valiron's theorem corresponds to Bloch's theorem. Based on his Principle
Bloch's_principle
language, the Picard group is infinite cyclic, other than for a short list of degrees. This is now often called the Noether-Lefschetz theorem. v t e
Max Noether's theorem on curves
Max_Noether's_theorem_on_curves
Theorem in complex analysis
analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic
Argument_principle
Index of articles associated with the same name
Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. Picard–Lindelöf theorem, the uniqueness of solutions
Uniqueness_theorem
Attribute of a mathematical function
allow the determination of general contour integrals via the residue theorem. The residue of a meromorphic function f {\displaystyle f} at an isolated
Residue_(complex_analysis)
Functions in mathematics
principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions
Harmonic_function
Undefined point on a holomorphic function which can be made regular
removable nor a pole, it is called an essential singularity. The Great Picard Theorem shows that such an f {\displaystyle f} maps every punctured open neighborhood
Removable_singularity
Complex-differentiable (mathematical) function
holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred
Holomorphic_function
Mathematical theorem in complex analysis
D} . This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to
Maximum_modulus_principle
less well-known identities can be deduced in a similar manner. The Picard theorem can be proved using the winding properties of planar Brownian motion
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Geometric representation of the complex numbers
giving a contour integral that is not necessarily zero, by the residue theorem. Cutting the complex plane ensures not only that Γ(z) is holomorphic in
Complex_plane
Theorem on holomorphic functions
In complex analysis, the open mapping theorem states that if U {\displaystyle U} is a domain of the complex plane C {\displaystyle \mathbb {C} } and f
Open mapping theorem (complex analysis)
Open_mapping_theorem_(complex_analysis)
Generalisation of Jacobian variety
_{0}V)^{\vee }.} For algebraic curves, the Abel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic. Intermediate Jacobian Albanese
Albanese_variety
Concept in complex analysis
of potential functions for conservative vector fields, in that Green's theorem is only able to guarantee path independence when the function in question
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Division of mathematical analysis
a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for
Value distribution theory of holomorphic functions
Value_distribution_theory_of_holomorphic_functions
Conformal mappings in complex analysis
Theorems Analyticity of holomorphic functions Cauchy's integral theorem Cauchy's integral formula Residue theorem Liouville's theorem Picard theorem Weierstrass
Schwarz_triangle_function
American mathematician (1928-1993)
1090/S0002-9939-1974-0338465-0. MR 0338465. Royden, H. L. (1984). "The Picard theorem for Riemann surfaces". Proc. Amer. Math. Soc. 90 (4): 571–574. doi:10
Halsey_Royden
Theorem
at the point and vice versa.) Among the corollaries of this theorem are the identity theorem that two holomorphic functions that agree at every point of
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Root-finding algorithm
whenever the real part of a {\displaystyle a} is negative. The Picard–Lindelöf theorem, which shows that ordinary differential equations have solutions
Fixed-point_iteration
References to the famous problem in number theory
Next Generation, begins with Picard attempting to solve the puzzle in his ready room; he remarks to Riker that the theorem had remained unproven for 800
Fermat's Last Theorem in fiction
Fermat's_Last_Theorem_in_fiction
French mathematician (1893–1948)
far-reaching generalization of Picard's theorem.) His proof of this theorem contained gaps (which he recognized), and later the theorem was known as "Bloch's conjecture"
André_Bloch_(mathematician)
Statement on solutions to ordinary differential equations
value problem. Mathematics portal Picard–Lindelöf theorem Cauchy–Kowalevski theorem Coddington & Levinson (1955), Theorem 1.2 of Chapter 1 Coddington & Levinson
Carathéodory's existence theorem
Carathéodory's_existence_theorem
The area cut off by a secant of a smooth convex oval is not an algebraic function
In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant
Newton's_theorem_about_ovals
Pseudometric of complex manifolds
C. Nevanlinna theory is a more quantitative descendant of Picard's theorem. Brody's theorem says that a compact complex space X is Kobayashi hyperbolic
Kobayashi_metric
Has no other singularities close to it
important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated. Isolated
Isolated_singularity
Mathematical functions which are smooth but not analytic
and hence is not even continuous, much less analytic. By the great Picard theorem, it attains every complex value (with the exception of zero) infinitely
Non-analytic_smooth_function
Infinite sum that is considered independently from any notion of convergence
for example, that its radius of convergence is 1 by the Cauchy–Hadamard theorem. However, as a formal power series, we may ignore this completely; all
Formal_power_series
Number not in the image of a given function
the only lacunary value of the complex exponential function. The two Picard theorems limit the number of possible lacunary values of certain types of holomorphic
Lacunary_value
Type of smooth complex surface of kodaira dimension 0
Serre's GAGA theorem. The Picard group of a K3 surface X is always a finitely generated free abelian group; its rank is called the Picard number ρ {\displaystyle
K3_surface
sheaf cohomology, and description in terms of the Picard functor, was given by David Mumford. The theorem states that for any complete varieties U {\displaystyle
Theorem_of_the_cube
American mathematician
from Princeton University under Phillip Griffiths with thesis Some Picard Theorems for Holomorphic Maps to Algebraic Varieties. In 1970/71, Green was
Mark_Lee_Green
Branch of ordinary differential equations
defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form
Floquet_theory
Type of calculus problem
{\displaystyle y(0)} and y ′ ( 0 ) {\displaystyle y'(0)} . The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if
Initial_value_problem
Theorem in complex analysis
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application
Borel–Carathéodory_theorem
Existence and uniqueness theorem for certain partial differential equations
the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential
Cauchy–Kovalevskaya_theorem
Mathematical theorem
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Class of continuous maps between Riemannian manifolds of the same dimension
mappings. Springer Verlag. D. Drasin; Pekka Pankka (2015). "Sharpness of Rickman's Picard theorem in all dimensions". Acta Math. Vol. 214. pp. 209–306.
Quasiregular_map
Swiss mathematician (1888–1977)
Zurich awarded him habilitation for a thesis on complex analysis and Picard's theorem. The examiner was Ernst Zermelo. Bernays was Privatdozent at the University
Paul_Bernays
Matrix used in complex analysis
Ω extending to a continuous function on the closure, then, by Stokes' theorem applied to the differential 1-form ω = h ( z ) d z , {\displaystyle \omega
Grunsky_matrix
Term in mathematics
degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety. The Jacobian variety is named after
Jacobian_variety
12th episode of the 2nd season of Star Trek: The Next Generation
Captain Picard and Commander Riker about Fermat's Last Theorem, which in the canon of the series had remained unsolved for 800 years (the theorem was proved
The_Royale
Class of problems for PDEs
zero means that the function itself is specified. The Cauchy–Kovalevskaya theorem, named in honor of Cauchy and Sofya Kovalevskaya, states: If all the functions
Cauchy_problem
American mathematician
doi:10.1007/BF02760922 with Pekka Pannka: "Sharpness of Rickman’s Picard theorem in all dimensions." Acta Mathematica 214, no. 2 (2015): 209–306. doi:10
David_Drasin
Polish mathematician (1892–1945)
Banach spaces. Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was
Stefan_Banach
Theorem in algebraic geometry
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected
Chevalley's_structure_theorem
Initial estimate or framework to the solution of a mathematical problem
results. An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution
Ansatz
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
differential equations, whose solvability is guaranteed by the Picard–Lindelöf theorem. If the vector field X {\displaystyle X} is nowhere zero then it
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Differential equation that is linear with respect to the unknown function
order at least two. This is the main result of Picard–Vessiot theory which was initiated by Émile Picard and Ernest Vessiot, and whose recent developments
Linear_differential_equation
Theorem in algebraic geometry
homology. In this setting, the theorem holds for highly singular spaces. A Lefschetz-type theorem also holds for Picard groups. Let X {\displaystyle X}
Lefschetz_hyperplane_theorem
Mathematics book
221]; V. Les schémas de Picard. Théorèmes d'existence [Séminaire Bourbaki, t. 14, 1961/62, no. 232]; VI. Les schémas de Picard. Propriétés générales [Séminaire
Fondements de la Géometrie Algébrique
Fondements_de_la_Géometrie_Algébrique
Generalized function whose value is zero everywhere except at zero
Joseph Fourier. Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur (1822) in the form: f
Dirac_delta_function
French mathematician (1886-1971)
while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent
Paul_Lévy_(mathematician)
Type of constraint on solutions to differential equations
uniqueness Well-posed problem Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kovalevskaya theorem General topics Initial
Dirichlet_boundary_condition
Class of ordinary differential equations
differential equation can be solved using ordinary methods and the Picard–Lindelöf theorem ensures that the differential equation has a unique solution in
Sturm–Liouville_theory
American mathematician
under Phillip Griffiths. His doctoral dissertation was titled, Picard–Lefschetz Theorem for Families of Algebraic Varieties Acquiring Certain Singularities
Herbert_Clemens
Theorem in the representation theory of linear algebraic groups
[1994], "Lie-Kolchin theorem", Encyclopedia of Mathematics, EMS Press Kolchin, E. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous
Lie–Kolchin_theorem
Property of differential equations describing physical phenomena
There are many results on this topic. For example, the Cauchy–Kowalevski theorem for Cauchy initial value problems essentially states that if the terms
Well-posed_problem
Group in algebraic geometry
Néron–Severi theorem, which was proved by Severi over the complex numbers and by Néron over more general fields. In other words, the Picard group fits into
Néron–Severi_group
Differential equations involving stochastic processes
equation and is defined on a given probability space. The Yamada–Watanabe theorem makes a connection between the two. An important example is the equation
Stochastic differential equation
Stochastic_differential_equation
Type of boundary condition in mathematics
uniqueness Well-posed problem Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kovalevskaya theorem General topics Initial
Robin_boundary_condition
German mathematician (1843–1921)
work on the latter allowed Émile Picard to show solutions of differential equations exist (the Picard–Lindelöf theorem). In 1892 he became a member of
Hermann_Schwarz
Method for solving continuous operator problems (such as differential equations)
c\|u\|^{2}} for some constant c > 0. {\displaystyle c>0.} By the Lax-Milgram theorem (see weak formulation), these two conditions imply well-posedness of the
Galerkin_method
Projective variety that is also an algebraic group
appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group
Abelian_variety
Methods of calculating definite integrals
C 1 ( [ a , b ] ) . {\displaystyle f\in C^{1}([a,b]).} The mean value theorem for f , {\displaystyle f,} where x ∈ [ a , b ) , {\displaystyle x\in [a
Numerical_integration
Type of functional equation (mathematics)
equations Numerical methods for partial differential equations Picard–Lindelöf theorem on existence and uniqueness of solutions Recurrence relation, also
Differential_equation
Lindelöf's theorem, a mathematical theorem within complex analysis Phragmén–Lindelöf principle, a mathematical principle Picard–Lindelöf theorem, a mathematical
Lindelöf_(surname)
PICARD THEOREM
PICARD THEOREM
Boy/Male
British, Danish, English, French, Irish
Strong Ruler; Powerful Leader; Rich and Powerful Ruler
Surname or Lastname
English
English : from Middle English pa(c)k ‘pack’, ‘bundle’ + the Anglo-Norman French pejorative suffix -ard, hence a derogatory occupational name for a peddler.English : pejorative derivative of the Middle English personal name Pack.English : from a Norman personal name, Pachard, Baghard, composed of the Germanic elements pac, bag ‘fight’ + hard ‘hardy’, ‘brave’, ‘strong’.Probably an Americanized spelling of German Packert, Päckert, from Germanic personal names formed with a word meaning ‘battle’ or ‘to fight’; or a variant of Packer 2 (with excrescent -t).
Surname or Lastname
English (Devon and Cornwall) and German
English (Devon and Cornwall) and German : variant of Richard.Americanized spelling of German Reichardt.
Surname or Lastname
English
English : unexplained; probably of French origin (see 2).French : unflattering nickname from a derivative of Old French pite ‘pitiful’, ‘lamentable’, perhaps applied to a family living in extreme poverty.
Surname or Lastname
English, French, German, and Dutch
English, French, German, and Dutch : from a Germanic personal name
composed of the elements rīc ‘power(ful)’ + hard
‘hardy’, ‘brave’, ‘strong’.A Richard from Normandy is documented in Quebec City in 1669, with
the secondary surname
Male
Slovene
Slovene form of Old High German Ricohard, RIHARD means "powerful ruler."
Male
English
Middle English form of Anglo-Saxon Siweard, SIWARD means "sea-guard."
Male
English
English form of Norman French Richaud, RICHARD means "powerful ruler."
Surname or Lastname
English (mainly Yorkshire) and German
English (mainly Yorkshire) and German : variant of Picard.English : some early examples, such as Paganus filius Pichardi (Hampshire, 1160), seem to point to derivation from a Germanic personal name, probably composed of the elements bic ‘sharp point’, ‘pointed weapon’ + hard ‘hardy’, ‘brave’, ‘strong’.Dutch : regional name for someone from Picardy in northern France.German : variant of Picker 4.
Surname or Lastname
English
English : variant spelling of Vickery.
Surname or Lastname
English and French
English and French : variant of Richard.A Ricard is documented in Montreal in 1665, with the secondary surname Saint-Germain.
Male
Spanish
Spanish form of Latin Ricardus, RICARDO means "powerful ruler."
Surname or Lastname
English
English : metonymic occupational name for someone who carded wool (i.e. disentangled it), preparatory to spinning, from Middle English, Old French card(e) ‘carder’, an implement used for this purpose.Reduced form of Irish McCard.
Boy/Male
Shakespearean
The Tragedy of Macbeth' Siward, Earl of Northumberland, general of the English forces. Also Young...
Male
Scandinavian
Scandinavian form of Old High German Ricohard, RIKARD means "powerful ruler."
Male
English
Variant spelling of English Giffard, GIFARD means "chubby-cheeked."
Boy/Male
French
Powerful; strong ruler.
Female
Spanish
Feminine form of Spanish Ricardo, RICARDA means "powerful ruler." Used mostly in Germany.
Female
Turkish
Turkish name PINAR means "spring."
Surname or Lastname
English
English : variant spelling of Vickers.
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