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Identity for a sequence of orthogonal polynomials
In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin
Christoffel–Darboux_formula
Summation formula
In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux (1876) for summing infinite series by using integrals or evaluating
Darboux's_formula
French mathematician (1842–1917)
theorem Darboux's formula Christoffel–Darboux identity Christoffel–Darboux formula Euler–Darboux equation Darboux–Froda's theorem Euler–Poisson–Darboux equation
Jean_Gaston_Darboux
Summation formula
Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula. If m and n are natural numbers and f(x) is a real or complex valued
Euler–Maclaurin_formula
Polynomial sequence
P_{n}^{(\alpha ,\beta )}} for large n {\displaystyle n} is given by the Darboux formula P n ( α , β ) ( cos θ ) = n − 1 2 k ( θ ) cos ( N θ + γ ) + O (
Jacobi_polynomials
Set of polynomials where any two are orthogonal to each other
polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century
Orthogonal_polynomials
Mathematical method
In mathematics, the Darboux transformation, named after Gaston Darboux (1842–1917), is a method of generating a new equation and its solution from the
Darboux_transformation
Formulas in differential geometry
motion. Roughly speaking, the Frenet–Serret formulas express the Darboux derivative of the TNB frame. If the Darboux derivatives of two frames are equal, then
Frenet–Serret_formulas
Natural moving frame in differential geometry of surfaces
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame
Darboux_frame
German mathematician and physicist (1829–1900)
he also introduced the Christoffel–Darboux formula for Legendre polynomials (he later also published the formula for general orthogonal polynomials)
Elwin_Bruno_Christoffel
Polynomial sequence
T(n)={\frac {\operatorname {He} _{n}(i)}{i^{n}}}.} The Christoffel–Darboux formula for Hermite polynomials reads ∑ k = 0 n H k ( x ) H k ( y ) k ! 2 k
Hermite_polynomials
Random matrix with gaussian entries
x'):=\sum _{n=0}^{N-1}\psi _{n}(x)\psi _{n}(x')} , and by the Christoffel–Darboux formula, K N ( x , x ′ ) = e − 1 4 ( x 2 + x ′ 2 ) ( N − 1 ) ! 2 π He N (
Gaussian_ensemble
Sequence of differential equation solutions
symmetric kernel polynomial has the representations (Christoffel–Darboux formula)[citation needed] K n ( α ) ( x , y ) := 1 Γ ( α + 1 ) ∑ i = 0 n L
Laguerre_polynomials
three term recurrence or the Christoffel-Darboux formula hold. There exist however other recursion formulas for certain types of measures. There exist
Sobolev orthogonal polynomials
Sobolev_orthogonal_polynomials
Name list
equation, Christoffel symbols, Schwarz–Christoffel mapping, Christoffel–Darboux formula Louis Christoffel (1886–?), Belgian wrestler Martin Christoffel (1922–2001)
Christoffel
Formula describing the asymptotic behavior of the Legendre polynomials
polynomials, which are also called the Mehler–Heine formula. The formula complements the Darboux formulae which describe the asymptotics in the interior
Mehler–Heine_formula
Mathematical theorem, used in calculus
mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle
Integral_of_inverse_functions
these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Operation in calculus
Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD), derived a formula for the sum of fourth powers. Alhazen determined the equations to calculate
Integral
Partial differential equations with data on two intersecting characteristics
The Goursat problem (also called the Darboux problem) is a boundary value problem for a second-order hyperbolic partial differential equation (PDE) in
Goursat_problem
Basic integral in elementary calculus
the Darboux integral. First, one shows that the second definition is equivalent to the definition of the Darboux integral; for this see the Darboux integral
Riemann_integral
Approximation technique in integral calculus
x_{i}} ) is contained between the lower and upper Darboux sums. This forms the basis of the Darboux integral, which is ultimately equivalent to the Riemann
Riemann_sum
Mehler, student of Dirichlet (Ferdinand): Mehler's formula, Mehler–Fock formula, Mehler–Heine formula, Mehler functions Meijer G-function Josef Meixner:
List of eponyms of special functions
List_of_eponyms_of_special_functions
Trick relating differential forms
is the standard argument for the modern proof of Darboux's theorem, as well as for the proof of Darboux-Weinstein theorem and other normal form results
Moser's_trick
Generalization of the Riemann integral
{E} [f(X)]=\int _{-\infty }^{\infty }f(x)g'(x)\,\mathrm {d} x.} But this formula does not work if X does not have a probability density function with respect
Riemann–Stieltjes_integral
Field of combinatorics using complex analysis
circle method. Walter Hayman's 1956 paper "A Generalisation of Stirling's Formula" is considered one of the earliest examples of the saddle-point method
Analytic_combinatorics
Problem in physics and astronomy
centers of force. Special cases of these generalized problems include Darboux's problem and Velde's problem. Euler's three-body problem is to describe
Euler's_three-body_problem
Fundamental formulas linking the metric and curvature tensor of a manifold
Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas that link together the induced metric and second fundamental
Gauss–Codazzi_equations
Mathematical measure of how much a curve or surface deviates from flatness
denominator can equally well be taken to be d(P,Q)3. The formula is valid in any dimension. The formula follows by verifying it for the osculating circle. There
Curvature
equation Poisson kernel Poisson integral formula Poisson–Jensen formula Fourier analysis Poisson summation formula (Poisson resummation) Wavelet theory Poisson
List of things named after Siméon Denis Poisson
List_of_things_named_after_Siméon_Denis_Poisson
Generalization of an ordered basis of a vector space
frame (aircraft principal axes) when described by the pilot. Darboux frame Frenet–Serret formulas Turtle graphics Yaw, pitch, and roll Rotating reference frame
Moving_frame
Example of a phase-space star product in mathematics
arbitrary Poisson manifolds (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula. A simple explicit example of the construction
Moyal_product
Type of Diophantine equation
of convergents for the square root of seven are Applying the recurrence formula to this solution generates the infinite sequence of solutions (1, 0); (8
Pell's_equation
French mathematician (1856–1941)
Picard modular surface Picard theorem Picard variety Picard–Lefschetz formula Picard–Lindelöf theorem Picard–Vessiot theory Picard–Fuchs equation Painlevé
Émile_Picard
Coordinate system based on distances from a triangle's sidelines
are given by a' = kx, b' = ky, c' = kz where k can be determined by the formula k = 2 Δ a x + b y + c z {\displaystyle k={\tfrac {2\Delta }{ax+by+cz}}}
Trilinear_coordinates
Norwegian mathematician (1842–1899)
joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian)
Sophus_Lie
Common integrals in quantum field theory are set of formulas that are useful for computation of various types in quantum field theory such as partition
Common integrals in quantum field theory
Common_integrals_in_quantum_field_theory
British mathematician (1826–1883)
elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass formula in number theory. In matrix theory he is visible today in having his name
Henry_John_Stephen_Smith
PDE to describe nonlinear wave motion
parameter |citeseerx= (help) Nakamura, A. (1989). "A bilinear N-soliton formula for the KP equation". Journal of the Physical Society of Japan. 58 (2):
Kadomtsev–Petviashvili equation
Kadomtsev–Petviashvili_equation
Concept in mathematical analysis
kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over
Improper_integral
calculus) Cousin's lemma (real analysis) Danskin's theorem (convex analysis) Darboux's theorem (real analysis) Denjoy–Carleman theorem (functional analysis)
List_of_theorems
distance – Frederick J. Damerau and Vladimir Levenshtein Darboux function – Jean Gaston Darboux Darcy's law – Henry Darcy Darlington pair – Sidney Darlington
Scientific phenomena named after people
Scientific_phenomena_named_after_people
summation of products of into other summations Cesàro mean Abel's summation formula Convolution Cauchy product –is the discrete convolution of two sequences
List_of_real_analysis_topics
French mathematician (1858–1936)
the honeycombs are four-dimensional Euclidean polytopes. He derived a formula for the general displacement in four dimensions preserving the origin,
Édouard_Goursat
Group in group theory and physics
\mathbf {R} \to H(V)\to V\to 0.} Any symplectic vector space admits a Darboux basis {ej, fk}1 ≤ j,k ≤ n satisfying ω(ej, fk) = δjk and where 2n is the
Heisenberg_group
Decomposition of periodic functions
bodies, presented on December 21, 1807 to the National Institute]. In Darboux, Gaston (ed.). Oeuvres de Fourier [The Works of Fourier] (in French). Vol
Fourier_series
Uniform restraint of the change in functions
function is the proof of the inverse Fourier transformation formula. We first prove that the formula is true for test functions, there are densely many of them
Uniform_continuity
Mathematical transformation
Müller (1948) for analogous formulas in different notation). Laguerre's inversion formulas from 1882 (equivalent to those of Darboux in 1887) read: D ′ = D
Spherical_wave_transformation
geometry and list of Lie group topics. List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius of curvature
List of differential geometry topics
List_of_differential_geometry_topics
French mathematician, physicist and engineer (1854–1912)
way of thinking to how he made several discoveries. The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated
Henri_Poincaré
Norwegian geophysicist and mathematician
realium in 1898. He then studied with Picard, Poincaré, Painlevé, Jordan, Darboux, and Goursat at the Sorbonne in Paris from 1898 to 1900. He returned to
Carl_Størmer
Algebra associated to any vector space
sides. The area of this parallelogram is given by the standard determinant formula: Area = | det [ v w ] | = | det [ a c b d ] | = | a d − b c | . {\displaystyle
Exterior_algebra
Construct all metric spaces where lines resemble those on a sphere
case of the Euclidean plane, we come to the problem posed by Jean Gaston Darboux: "To determine all the calculus of variation problems in the plane whose
Hilbert's_fourth_problem
Algebraic structure designed for geometry
transflections, and point reflections. PGA allows projection, meet, and angle formulas to be derived from G ( 3 , 0 , 1 ) {\displaystyle {\mathcal {G}}(3,0,1)}
Geometric_algebra
French mathematician (1793–1880)
coniques, Gauthier-Villars, from University of Michigan. Chasles–Cayley–Brill formula Chasles's theorem (disambiguation) Asteroid 18510 Chasles Michel Chasles
Michel_Chasles
Type of number sequence
denote our sequence by aj = jα (where j starts from 0, to simplify the formula later). Let ℓ ≠ 0 be an integer. Since α is irrational, ℓα can never be
Equidistributed_sequence
Mathematics of real numbers and real functions
series). One theorem that is sometimes shown at the elementary level is Darboux's theorem, that the derivative of a differentiable function satisfies the
Real_analysis
109–114. CiteSeerX 10.1.1.422.7691. doi:10.1016/j.jqsrt.2003.10.001. "Euler-Darboux Equation". MathWorld. Retrieved 2009-11-14. Maslov, V. P. (1983). "The
List of scientific equations named after people
List_of_scientific_equations_named_after_people
Shortest paths on a bounded deformed sphere-like quadric surface
simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation
Geodesics_on_an_ellipsoid
Foundational law of electromagnetism relating electric field and charge distributions
Law", too. Lagrange, Joseph-Louis (1869) [1776]. Serret, Joseph-Alfred; Darboux, Jean-Gaston (eds.). "Sur l'attraction des sphéroïdes elliptiques" [On
Gauss's_law
Differential algebra
{\displaystyle V} , and replacing i ℏ {\displaystyle i\hbar } in the Moyal product formula with 1 {\displaystyle 1} .) The isomorphism is given by the symmetrization
Weyl_algebra
Branch of mathematics
realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature, is
Differential_geometry
Type of manifold in differential geometry
{\displaystyle \omega } along V H {\displaystyle V_{H}} vanishes. Applying Cartan's formula, this amounts to (here ι X {\displaystyle \iota _{X}} is the interior product):
Symplectic_manifold
"d'Alembert's Equation". mathworld.wolfram.com. Retrieved 2024-06-02. "Darboux equation - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved
List of nonlinear ordinary differential equations
List_of_nonlinear_ordinary_differential_equations
Jacobi identity Universal enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop
List_of_Lie_groups_topics
Dutch mathematician (1856–1894)
Stieltjes polynomials Stieltjes transformation (and Stieltjes inversion formula) Stieltjes–Wigert polynomials "Stieltjes integral". Oxford English Dictionary
Thomas_Joannes_Stieltjes
Mathematical concept
Maurer–Cartan form ω is a g-valued one-form on G defined on vectors v ∈ TgG by the formula ω g ( v ) = ( L g − 1 ) ∗ v . {\displaystyle \omega _{g}(v)=(L_{g^{-1}})_{*}v
Maurer–Cartan_form
Operation in Hamiltonian mechanics
for any f {\displaystyle f} . In canonical coordinates (also known as Darboux coordinates) ( q i , p i ) {\displaystyle (q_{i},\,p_{i})} on the phase
Poisson_bracket
Ramsey–Cass–Koopmans model Dynamic stochastic general equilibrium Feynman–Kac formula Black–Scholes equation Affine term structure modeling Fokker–Planck equation
List of named differential equations
List_of_named_differential_equations
Theorem in mathematics
that it must be entirely of one sign (positive or negative) according to Darboux's theorem, and therefore the function must be strictly monotone, and thus
Inverse_function_theorem
Branch of geometry
\alpha =dz-\Sigma _{i=1}^{n}y_{i}dx_{i}} . Such coordinates are called Darboux coordinates. In this sense, contact geometry is a stable distribution,
Contact_geometry
Charlois, astronomer, who discovered around 99 asteroids. Jean Gaston Darboux, mathematician, he made several important contributions to geometry and
List_of_Occitans
Indefinite integral
parts (to integrate products of functions) Inverse function integration (a formula that expresses the antiderivative of the inverse f−1 of an invertible and
Antiderivative
triangle. This can be proven by the midpoint theorem of triangles and Heron's formula. The orthocenter of the medial triangle coincides with the circumcenter
Midpoint_polygon
Differential equation containing derivatives with respect to only one variable
ISBN 0-471-83824-1 Boscain; Chitour 2011, p. 21 Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M. R. Spiegel, J. Liu, Schaum's
Ordinary differential equation
Ordinary_differential_equation
Space of all possible states that a system can take
local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space
Phase_space
Method of mathematical differentiation
{a}{b}}\right)=\ln(a)-\ln(b),\qquad \ln(a^{n})=n\ln(a).} Using Faà di Bruno's formula, the n-th order logarithmic derivative is, d n d x n ln f ( x ) = ∑ m
Logarithmic_differentiation
Type of center of a polygon
{\displaystyle CCM(P)} of the polygon P {\displaystyle P} is given by the formula C C M ( P ) = 1 4 A ( ∑ i = 0 n − 1 − y i y i + 1 2 + y i 2 y i + 1 + x
Circumcenter_of_mass
Method of mathematical integration
irrationals are both dense in the reals. Thus the upper Darboux sums are all one, and the lower Darboux sums are all zero. 1 Q {\displaystyle 1_{\mathbf {Q}
Lebesgue_integral
General form of integrable system
wave solutions, exact solitary wave solutions via bilinearization, and Darboux transformation together with the orbital stability for such solitary wave
Kundu_equation
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
a particle in a subset of 3D space, but we do not know its trajectory formula. Instead, we know only that its trajectory satisfies a d x + b d y + c
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Matrix-valued random variable
explicitly computed, see Selberg integral. In the case of GUE (β = 2), the formula (1) describes a determinantal point process. Eigenvalues repel as the joint
Random_matrix
Geometric inversion of a torus, cylinder or double cone
An elliptic cyclide can be represented parametrically by the following formulas (see section Cyclide as channel surface): x = d ( c − a cos u cos v
Dupin_cyclide
compose a problem, which is solved using the Riemann–Volterra integral formula. This yields the generic solution expressed via a double integral over
Spacetime triangle diagram technique
Spacetime_triangle_diagram_technique
Surface that locally minimizes its area
Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by Heinrich
Minimal_surface
an important issue in the theory of integrability, due to the Laplace-Darboux transformations, which allow construction of integrable LPDEs. Laplace
Invariant factorization of LPDOs
Invariant_factorization_of_LPDOs
Vector calculus construction
_{t}\right)\right)_{p}} This last identity is useful to prove the Darboux theorem. Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag
Time_dependent_vector_field
Mathematics of smooth surfaces
Curvature of general surfaces was first studied by Euler. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered
Differential geometry of surfaces
Differential_geometry_of_surfaces
Mathematical structure in differential geometry
canonical coordinates coincides with the one described above. In general, by Darboux theorem, any arbitrary symplectic manifold ( M , ω ) {\displaystyle (M
Poisson_manifold
Development of linear transformations forming the Lorentz group
Ossian Bonnet (1856), Albert Ribaucour (1870), Sophus Lie (1871a), Gaston Darboux (1873-87), Edmond Laguerre (1880), Cyparissos Stephanos (1883), Georg Scheffers
History of Lorentz transformations
History_of_Lorentz_transformations
about the depth of the ideas presented by Riemann." Gaston Darboux Publication data: Darboux, Gaston (1890). Leçons sur la théorie génerale des surfaces
List of publications in mathematics
List_of_publications_in_mathematics
are linearly independent. In the special case of a graph y = y(x), these formulas reduce to k = − 1 2 ( 1 ( y ″ ) 2 3 ) ″ = y ⁗ 3 ( y ″ ) 5 3 − 5 ( y ‴ )
Affine_curvature
Intrinsic geometric structures in mathematics
France by Darboux and Goursat. It also echoed parallel developments in Albert Einstein's theory of relativity. Objects appearing in the formulas of Gauss
Riemannian connection on a surface
Riemannian_connection_on_a_surface
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA
DARBOUXS FORMULA