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Solutions of Legendre's differential equation
science and mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμ λ, Qμ λ, and Legendre functions of the second kind, Qn, are
Legendre_function
System of complete and orthogonal polynomials
related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials
Legendre_polynomials
Canonical solutions of the general Legendre equation
complex values of ℓ and m are Legendre functions. In that case the parameters are usually labelled with Greek letters. The Legendre ordinary differential equation
Associated Legendre polynomials
Associated_Legendre_polynomials
Mathematical transformation
variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, the Legendre transform is used
Legendre_transformation
Bessel–Clifford function Kelvin functions Legendre function: From the theory of spherical harmonics. Scorer's function Sinc function Hermite polynomials
List of mathematical functions
List_of_mathematical_functions
Function in number theory
In number theory, the Legendre symbol is a function of a {\displaystyle a} and p {\displaystyle p} defined as ( a p ) = { 1 if a is a quadratic residue
Legendre_symbol
Mathematical function
In mathematics, the Legendre chi function (named after Adrien-Marie Legendre) is a special function whose Taylor series is also a Dirichlet series, given
Legendre_chi_function
French mathematician (1752–1833)
Adrien-Marie Legendre (/ləˈʒɑːndər, -ˈʒɑːnd/; French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous
Adrien-Marie_Legendre
Special mathematical functions defined on the surface of a sphere
between the vectors x and x1. The functions P i : [ − 1 , 1 ] → R {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } are the Legendre polynomials, and they can be
Spherical_harmonics
Type of function in mathematics
the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials
Legendre_rational_functions
C standard library header file
operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. The C++ standard library
C_mathematical_functions
Generalization of the Legendre transformation
conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation
Convex_conjugate
Numerical analysis concept
numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over
Gauss–Legendre_quadrature
Extension of the factorial function
algebra. The name gamma function and the symbol Γ {\displaystyle \Gamma } were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's
Gamma_function
Class of periodic mathematical functions
{\displaystyle \wp } -function The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work
Elliptic_function
Solutions of the Helmholtz equation
harmonics. Both type of spheroidal harmonics are expressible in terms of Legendre functions. Oblate spheroidal coordinates, especially the section Oblate spheroidal
Spheroidal_wave_function
Mathematical function
0 {\displaystyle \Re (z_{2})>0} . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its
Beta_function
−v;1+μ;1−x2)Γ(1+μ)) Legendre function Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Ferrers Function", NIST Handbook
Ferrers_function
Gauss–Legendre algorithm Gauss–Legendre method Gauss–Legendre quadrature Legendre (crater) Legendre chi function Legendre duplication formula Legendre–Papoulis
List of things named after Adrien-Marie Legendre
List_of_things_named_after_Adrien-Marie_Legendre
as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f ( x ) {\displaystyle f(x)}
Legendre transform (integral transform)
Legendre_transform_(integral_transform)
Mathematical description of quantum state
integrable functions on the unit sphere S2 is a Hilbert space. The basis functions in this case are the spherical harmonics. The Legendre polynomials
Wave_function
Function defined by a hypergeometric series
include most of the commonly used functions of mathematical physics. Legendre functions are solutions of a second order differential equation with 3 regular
Hypergeometric_function
Type of function
in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential equations
Orthogonal_functions
Product of numbers from 1 to n
continuous extension of the factorial function to the gamma function. Adrien-Marie Legendre included Legendre's formula, describing the exponents in the
Factorial
Type of wavelet
supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications
Legendre_wavelet
Nearest integers from a number
the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket
Floor_and_ceiling_functions
In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression
Whipple_formulae
Polynomial sequence
on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special
Gegenbauer_polynomials
solutions of a differential equation) is a constant. Legendre's relation stated using elliptic functions is ω 2 η 1 − ω 1 η 2 = 2 π i {\displaystyle \omega
Legendre's_relation
Special function defined by an integral
rational functions and the three Legendre canonical forms, also known as the elliptic integrals of the first, second and third kind. Besides the Legendre form
Elliptic_integral
German mathematician (1821–1881)
on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions (Handbuch der
Eduard_Heine
Number theory expression
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after
Legendre's_formula
Second-order partial differential equation
} Here Yℓm is called a spherical harmonic function of degree ℓ and order m, Pℓm is an associated Legendre polynomial, N is a normalization constant,
Laplace's_equation
Function representing the number of primes less than or equal to a given number
growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately xlogx where
Prime-counting_function
Element of a basis for a function space
and Legendre polynomials. Different polynomial bases can have different numerical properties, particularly when the degree is large. Basis functions can
Basis_function
Identity obeyed by many special functions related to the gamma function
}}\;\Gamma (2z).} It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is Γ
Multiplication_theorem
Mathematical functions
arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form: These functions can be displayed directly by using the incomplete elliptic integral
Lemniscate_elliptic_functions
Liquid shape formed in electrospraying
2 ( cos θ 0 ) {\displaystyle P_{1/2}(\cos \theta _{0})\,} (the Legendre function of order 1/2). Taylor's derivation is based on two assumptions: (1)
Taylor_cone
Partial differential equations
three-variable Laplace equation, is given in terms of the generating function for Legendre polynomials, 1 | x − x ′ | = ∑ l = 0 ∞ r < l r > l + 1 P l ( cos
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum. Conversely, the inverse
Finite_Legendre_transform
French polymath (1749–1827)
sequence of functions P0k(cos φ) is the set of so-called "associated Legendre functions" and their usefulness arises from the fact that every function of the
Pierre-Simon_Laplace
Special type of functions in mathematics
associated Legendre polynomials. For c ≠ 0 {\displaystyle c\neq 0} , the angular spheroidal wave functions can be expanded as a series of Legendre functions. If
Prolate spheroidal wave function
Prolate_spheroidal_wave_function
Operation in calculus
antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending
Integral
Says when a natural number is the sum of three squares of integers
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers n=x2+y2+z2 if and
Legendre's three-square theorem
Legendre's_three-square_theorem
Formula for the derivative of an inverse function
f(x) is also of interest, as it is used in showing the convexity of the Legendre transform. Let z=f′(x) then we have, assuming f″(x)≠0:ddz[f′]−1(z)=1f″(x)This
Inverse_function_rule
Fourier expansion of a reciprocal square root
{1}{2}}}(z)e^{im\psi },} where Q m − 1 / 2 {\displaystyle Q_{m-1/2}} is a Legendre function of the second kind and z ∈ R > 1 {\displaystyle z\in \mathbb {R} _{>1}}
Heine's_identity
Quickly converging computation of π
The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing
Gauss–Legendre_algorithm
Integral expressing the amount of overlap of one function as it is shifted over another
mathematical operation on two functions f and g that produces a third function f∗g, as the integral of the product of the two functions after one is reflected
Convolution
Smoothed ramp function
the softplus function is the negative binary entropy function (with base e). This is because (following the definition of the Legendre transformation:
Softplus
Special function in mathematics
discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function. The values of ζ(s, a) at s = 0, −1, −2,
Hurwitz_zeta_function
Mathematical function
In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind
Conical_function
Approximation of the definite integral of a function
degree 2n − 1 or less on [−1, 1]. The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if
Gaussian_quadrature
Constant of proportionality of prime number density
in the third and fourth columns are estimated using the Riemann R function. Legendre, A.-M. (1808). Essai sur la théorie des nombres [Essay on number theory]
Legendre's_constant
Distance from origin of tangent hyperplanes
using the fact that the Legendre transform of a positive homogeneous, convex, real valued function is the (convex) indicator function of a compact convex
Support_function
(1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. (Doklady) Acad. Sci
Mehler–Fock_transform
Mathematical concept
In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve
Legendre_sieve
Lambert W function Gabriel Lamé: Lamé polynomial G. Lauricella Lauricella-Saran: Lauricella hypergeometric series Adrien-Marie Legendre: Legendre polynomials
List of eponyms of special functions
List_of_eponyms_of_special_functions
Family of numerical methods
Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. All Gauss–Legendre methods are A-stable. The Gauss–Legendre method
Gauss–Legendre_method
Three-dimensional orthogonal coordinate system
\phi }} Where P and Q are associated Legendre functions of the first and second kind. These Legendre functions are often referred to as toroidal harmonics
Toroidal_coordinates
In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image
Legendre_moment
Special type of functions in mathematics
the associated Legendre polynomials. For c≠0, the angular spheroidal wave functions can be expanded as a series of Legendre functions. Such expansions
Oblate spheroidal wave function
Oblate_spheroidal_wave_function
Gives conditions for the solvability of quadratic equations modulo prime numbers
product of the Riemann zeta function and a certain Dirichlet L-function The Jacobi symbol is a generalization of the Legendre symbol; the main difference
Quadratic_reciprocity
Family of power series in mathematics
Legendre polynomials and Chebyshev polynomials. A wide range of integrals of elementary functions can be expressed using the hypergeometric function,
Generalized hypergeometric function
Generalized_hypergeometric_function
Smallest convex set containing a given set
convex hull or lower convex envelope of a function f {\displaystyle f} on a real vector space is the function whose epigraph is the lower convex hull of
Convex_hull
Derivative of a function with multiple variables
differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced
Partial_derivative
Branch of mathematics
André (1984). Number theory: An approach through History from Hammurapi to Legendre. Boston: Birkhauser Boston. p. 28. ISBN 0-8176-4565-9. Hollingdale, Stuart
Calculus
Standard model of the structure of Earth's magnetic field
P_{n}^{m}\left(\cos \theta \right)} are the Schmidt quasi-normalized associated Legendre functions of degree n {\displaystyle n} and order m {\displaystyle m} P n m
International Geomagnetic Reference Field
International_Geomagnetic_Reference_Field
Function whose domain is the positive integers
theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range
Arithmetic_function
Textbook in mathematical analysis
Transcendental Functions The Gamma Function The Zeta Function of Riemann The Hypergeometric Function Legendre Functions The Confluent Hypergeometric Function Bessel
A_Course_of_Modern_Analysis
Document that proposed additions to the C++ standard library
C++11. additions to the <cmath>/<math.h> header files – beta, legendre, etc. These functions will likely be of principal interest to programmers in the engineering
C++_Technical_Report_1
Function in harmonic analysis on groups
hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by where α = ρ(ρ+1). Further restrictions
Zonal_spherical_function
1090/proc/13078, S2CID 119721248 Jackson, F. H. (1906a), "I.—On generalized functions of Legendre and Bessel", Transactions of the Royal Society of Edinburgh, 41
Jackson_q-Bessel_function
Smooth nowhere-analytic function
In mathematics, the Fabius function is a smooth cumulative distribution function F : R → [ 0 , 1 ] {\displaystyle F:\mathbb {R} \to [0,1]} that is strictly
Fabius_function
Mathematical function relating circular and hyperbolic functions
{gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s
Gudermannian_function
Formulation of classical mechanics using momenta
{q}}^{i}}}-{\mathcal {L}}.} The Legendre transform of L {\displaystyle {\mathcal {L}}} turns E L {\displaystyle E_{\mathcal {L}}} into a function H ( p , q , t ) {\displaystyle
Hamiltonian_mechanics
{\hbox{with}}\quad h:={\frac {r'}{r}}.} We find here the generating function of the Legendre polynomials P ℓ ( cos γ ) {\displaystyle P_{\ell }(\cos \gamma
Laplace_expansion_(potential)
Entropy of a process with only two probable values
Legendre transform) of the binary entropy (with base e) is the negative softplus function. This is because (following the definition of the Legendre transform:
Binary_entropy_function
Quantum mechanical potential
Thus the solutions ψ ( u ) {\displaystyle \psi (u)} are just the Legendre functions P λ μ ( tanh ( x ) ) {\displaystyle P_{\lambda }^{\mu }(\tanh(x))}
Pöschl–Teller_potential
Arithmetic function
Liouville function is a non-trivial example of a completely multiplicative function as are Dirichlet characters, the Jacobi symbol and the Legendre symbol
Completely multiplicative function
Completely_multiplicative_function
Irreducible representation of the rotation group SO
index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
Wigner_D-matrix
Operation on formal power series
other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include χ 1 ( z ) = ∑
Generating function transformation
Generating_function_transformation
Set of statistical processes for estimating the relationships among variables
time. The method of least squares was published by Legendre in 1805, and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of
Regression_analysis
Special mathematical function
{\operatorname {Ti} _{n}(t)}{t}}dt,} which explains the function name. The Legendre chi function χs(z) (Lewin 1958, Ch. VII § 1.1; Boersma & Dempsey 1992)
Polylogarithm
Canonical set of three elliptic integrals
mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the
Legendre_form
Type of artificial neural network
derivative of the activation function, and so this algorithm represents a backpropagation of the activation function. Circa 1800, Legendre (1805) and Gauss (1795)
Feedforward_neural_network
Mathematical concept
is the rate function in Sanov's theorem. Convex conjugate – Generalization of the Legendre transformation Integral of inverse functions – Mathematical
Young's inequality for products
Young's_inequality_for_products
polynomials q-Krawtchouk polynomials q-Laguerre polynomials Continuous q-Legendre polynomials q-Meixner polynomials q-Meixner–Pollaczek polynomials q-Racah
List_of_q-analogs
Mathematical series
The Cartesian approach has the advantage that no prior knowledge of Legendre functions, spherical harmonics, etc., is required. Its disadvantage is that
Multipole_expansion
Property of functions which is weaker than continuity
convex function. Some operations in convex analysis, such as the Legendre transform automatically produce closed convex functions. The Legendre transform
Semi-continuity
Mathematical function describing fluid motion
latitude and may be expressed as an infinite sum of associated Legendre polynomials; the functions are orthogonal over the sphere in the continuous case. Thus
Hough_function
Polynomial sequence
{\sigma -\tau }{\sqrt {2}}}.} Hermite transform Legendre polynomials Mehler kernel Parabolic cylinder function Romanovski polynomials Turán's inequalities
Hermite_polynomials
Symmetric holomorphic function
In mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the
Modular_lambda_function
Sequence of differential equation solutions
Physicists. Academic Press. ISBN 978-0-12-059825-0. Timothy Jones. "The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of
Laguerre_polynomials
Mathematical approximation of a function
_{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}} The Legendre chi functions are defined as follows: χ 2 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 x
Taylor_series
1964 mathematical reference work edited by M. Abramowitz and I. Stegun
Transcendental Functions Exponential Integral and Related Functions Gamma Function and Related Functions Error Function and Fresnel Integrals Legendre Functions Bessel
Abramowitz_and_Stegun
Approximation method in statistics
was published by Legendre in 1805. The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the
Least_squares
Function equal to the product of its values on coprime factors
Ramanujan tau function All Dirichlet characters are completely multiplicative functions, for example ( n / p ) {\displaystyle (n/p)} , the Legendre symbol,
Multiplicative_function
_{B}=\iota _{A\cap B}} . For an arbitrary non-empty set its Legendre transform is the support function. The subgradient of ι A ( x ) {\displaystyle \iota _{A}(x)}
Indicator function (convex analysis)
Indicator_function_(convex_analysis)
Lower bound on the quality factor of small radio antennae
being Legendre functions and spherical Bessel functions. The impedance could be expressed as a series of a ratio of a derivative of a Hankel function to
Chu–Harrington_limit
_{i=1}^{n}(\mu -\mu _{i})} where μ i {\displaystyle \mu _{i}} are the zeros of Legendre polynomials and W ( μ ) = ∏ α = 1 n ( 1 − k α 2 μ 2 ) {\displaystyle W(\mu
Chandrasekhar's X- and Y-function
Chandrasekhar's_X-_and_Y-function
travel, tourism, insurance
LEGENDRE FUNCTION
LEGENDRE FUNCTION
Boy/Male
Anglo Saxon
Legend name.
Boy/Male
French, German, Greek
Lion-man; Form of Leander; Brave as a Lion
Boy/Male
French
Form of Leander. 'Lionlike man.
Girl/Female
Arthurian Legend
In Arthurian legend Igrayne is mother of Arthur.
Boy/Male
Indian
The Legendary
Girl/Female
Arthurian Legend American French Greek
In Arthurian legend, Elaine was mother to Sir Lancelot's son Galahad.
Boy/Male
Celtic Arthurian Legend English Scottish Welsh
From Arthurian legend.
Boy/Male
Sikh
Source of light
Girl/Female
Celtic
Legendsry tale.
Girl/Female
French
Legend.
Boy/Male
Indian, Telugu
Legend
Boy/Male
Welsh
Legendary nobleman.
Girl/Female
Arthurian Legend English
Abbreviation of Lynnette who accompanied Sir Gareth on a knightly quest in Arthurian legend;Irish...
Boy/Male
Tamil
Jayachandra | ஜயசஂதà¯à®°Â
The legend
Jayachandra | ஜயசஂதà¯à®°Â
Girl/Female
English American Italian Latin
Legendary princess.
Boy/Male
Hindu
The legend
Girl/Female
Arthurian Legend
In Arthurian legend Igrayne is mother of Arthur.
Girl/Female
Australian, Celtic
Legendary Tale
Boy/Male
Arthurian Legend Welsh
Gentle. Modest and brave Sir Gareth was a legendary knight of King Arthur's Round Table.
Girl/Female
French
Legend.
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
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