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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
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2
Associated Legendre polynomials
Associated_Legendre_polynomials
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
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
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
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
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
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
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
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
Function defined by a hypergeometric series
functions. These include most of the commonly used functions of mathematical physics. Legendre functions are solutions of a second order differential equation
Hypergeometric_function
Mathematical function
(z_{1}),\operatorname {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
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
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
Type of function
in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential equations
Orthogonal_functions
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
C standard library header file
operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. Different C standards
C_mathematical_functions
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
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
Type of wavelet
supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications
Legendre_wavelet
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 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
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
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
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
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
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
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
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)
)}}\right)} Legendre function Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Ferrers Function", NIST Handbook
Ferrers_function
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 x log x {\displaystyle
Prime-counting_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
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
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
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
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 = x 2 + y 2 +
Legendre's three-square theorem
Legendre's_three-square_theorem
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
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
Fourier expansion of a reciprocal square root
{1}{2}}}(z)e^{im\psi }} where Q m − 1 2 {\displaystyle Q_{m-{\frac {1}{2}}}} is a Legendre function of the second kind, which has degree, m − 1⁄2, a half-integer, and
Heine's_identity
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
Operation in mathematical calculus
antiderivatives, the special functions (like the Legendre functions, the hypergeometric function, the gamma function, the incomplete gamma function and so on). Extending
Integral
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
Concept from evolutionary biology
membrane, was argued to be representable as a series of normalised Legendre functions. The algebraic solution of the above equations ran to some 30 pages
Turing_pattern
Natural number
value of Legendre's constant, introduced in 1808 by Adrien-Marie Legendre to express the asymptotic behavior of the prime-counting function. The Weil's
1
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
Integral expressing the amount of overlap of one function as it is shifted over another
a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle f*g} , as the
Convolution
Function whose domain is the positive integers
(n)}}.} In this formula ( a p ) {\displaystyle ({\tfrac {a}{p}})} is the Legendre symbol, defined for all integers a and all odd primes p by ( a p ) = {
Arithmetic_function
Mathematical function relating circular and hyperbolic functions
typo. Legendre (1817) §4.2.8(163) pp. 144–145 Kennelly (1929) p. 182 Kahlig & Reich (2013) Cayley (1862) p. 21 Kennelly (1929) pp. 180–183 Legendre (1817)
Gudermannian_function
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
Part of spectral theory
(1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. Acad. Sci. URSS, 39
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
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. Such
Oblate spheroidal wave function
Oblate_spheroidal_wave_function
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
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
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
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 ) {\displaystyle z=f'(x)} then we have, assuming
Inverse_function_rule
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
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
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
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
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
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum. Conversely, the inverse
Finite_Legendre_transform
Number, approximately 3.14
continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882
Pi
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
Polynomial sequence
{\sigma -\tau }{\sqrt {2}}}.} Hermite transform Legendre polynomials Mehler kernel Parabolic cylinder function Romanovski polynomials Turán's inequalities
Hermite_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
Thermodynamic potential
of interest given the partition function and are often used in density of state calculations. One can also do Legendre transformations for different systems
Helmholtz_free_energy
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
Symmetric holomorphic function
^{2}(1-\lambda )^{2}}}\ .} which is the j-invariant of the elliptic curve of Legendre form y 2 = x ( x − 1 ) ( x − λ ) {\displaystyle y^{2}=x(x-1)(x-\lambda
Modular_lambda_function
Subphylum of single-celled organisms
membrane, was argued to be representable as a series of normalised Legendre functions. The algebraic solution of the above equations ran to some 30 pages
Radiolaria
{\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)
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
polynomials q-Krawtchouk polynomials q-Laguerre polynomials Continuous q-Legendre polynomials q-Meixner polynomials q-Meixner–Pollaczek polynomials q-Racah
List_of_q-analogs
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
Physics of heat, work, and temperature
thermodynamic system. Other thermodynamic potentials can also be obtained through Legendre transformation. Axiomatic thermodynamics is a mathematical discipline that
Thermodynamics
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
Decompositions of inner product spaces into orthonormal bases
may be advantageous to use such Fourier–Legendre series rather than Fourier series since the basis functions for the series expansion are all polynomials
Generalized_Fourier_series
LEGENDRE FUNCTION
LEGENDRE FUNCTION
Girl/Female
English American Italian Latin
Legendary princess.
Girl/Female
Arthurian Legend
In Arthurian legend Igrayne is mother of Arthur.
Boy/Male
Tamil
Jayachandra | ஜயசஂதà¯à®°Â
The legend
Jayachandra | ஜயசஂதà¯à®°Â
Girl/Female
Arthurian Legend
In Arthurian legend Igrayne is mother of Arthur.
Girl/Female
French
Legend.
Boy/Male
French, German, Greek
Lion-man; Form of Leander; Brave as a Lion
Boy/Male
Indian, Telugu
Legend
Boy/Male
Sikh
Source of light
Boy/Male
Anglo Saxon
Legend name.
Girl/Female
Australian, Celtic
Legendary Tale
Boy/Male
Indian
The Legendary
Girl/Female
Arthurian Legend English
Abbreviation of Lynnette who accompanied Sir Gareth on a knightly quest in Arthurian legend;Irish...
Girl/Female
Arthurian Legend American French Greek
In Arthurian legend, Elaine was mother to Sir Lancelot's son Galahad.
Boy/Male
French
Form of Leander. 'Lionlike man.
Girl/Female
French
Legend.
Girl/Female
Celtic
Legendsry tale.
Boy/Male
Welsh
Legendary nobleman.
Boy/Male
Arthurian Legend Welsh
Gentle. Modest and brave Sir Gareth was a legendary knight of King Arthur's Round Table.
Boy/Male
Celtic Arthurian Legend English Scottish Welsh
From Arthurian legend.
Boy/Male
Hindu
The legend
LEGENDRE FUNCTION
LEGENDRE FUNCTION
Boy/Male
Hindu
Wise
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Against Desire
Boy/Male
Indian
Brave
Boy/Male
Hindu, Indian, Marathi
Lord Krishna
Girl/Female
Muslim/Islamic
Beautiful
Boy/Male
Indian
Never Destroy
Boy/Male
Hindu, Indian
Little; Freckled
Female
Chinese
orchid.
Boy/Male
Hindu, Indian, Punjabi, Sikh, Traditional
Superior; Warrior of the Kingdom; Brave King
Girl/Female
Hebrew
Beauty.
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
LEGENDRE FUNCTION
a.
Pertaining to deeds or feats of arms; legendary.
n.
An inscription, motto, or title, esp. one surrounding the field in a medal or coin, or placed upon an heraldic shield or beneath an engraving or illustration.
a.
Adhering to a fixed circle of legends; cyclic; hence, mean; inferior. See Cyclic poets, under Cyclic.
n.
One who relates legends.
n.
A story or legend abounding in miracles.
n.
One of a class of legendary heroes, who fought frenzied by intoxicating liquors, and naked, regardless of wounds.
v. i.
Alt. of Degener
n.
A brief poetical sentiment; hence, any brief sentiment, motto, or legend; especially, one inscribed on a ring.
v. t.
To tell or narrate, as a legend.
n.
Any wonderful story coming down from the past, but not verifiable by historical record; a myth; a fable.
n.
That which is appointed to be read; especially, a chronicle or register of the lives of saints, formerly read at matins, and in the refectories of religious houses.
n.
Any one of the legendary Greek heroes who sailed with Jason, in the Argo, in quest of the Golden Fleece.
n.
A story respecting saints; especially, one of a marvelous nature.
a.
Of or pertaining to a legend or to legends; consisting of legends; like a legend; fabulous.
n.
A book of legends; a tale or narrative.
n.
A Grecian legendary hero, son of Jupiter and Danae, who slew the Gorgon Medusa.
a.
Pertaining to, or treating of, institutions; as, institutional legends.
n.
A reciter of gests or legendary tales; a story-teller.
a.
Of or pertaining to, or characteristic of, Ossian, a legendary Erse or Celtic bard.