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Everything about Legendre function
Solutions of Legendre's differential equation
In physical science and mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμλ, Qμλ, and Legendre functions of the second kind, Qn, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Legendre functionA 5-minute orientation
- Skim a recent paper: Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and Legendre interval inclusion functionInfluential work since 2022
- Pick up a classic book: Legendre-Tau approximation for functional differential equationsThe most-reprinted book on the subject
- Read the landmark paper: Analytical results for a Bessel function times Legendre polynomials class integralsCited 58 times
- Go sideways: Associated Legendre polynomialsA closely related topic
- Go sideways: Confluent hypergeometric functionA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
- Legendre's differential equation
- Solutions of the differential equation
- Legendre functions of the second kind (Qn)
- Associated Legendre functions of the second kind
- Integral representations
- Legendre function as characters
- Singularities of Legendre functions of the first kind (Pλ) as a consequence of symmetry
🏆 Landmark papers (most cited)
Analytical results for a Bessel function times Legendre polynomials class integrals
The asymptotic expansion of legendre function of large degree and order
Transfer-function matrix identification in MIMO systems via shifted Legendre polynomials
Shifted Legendre function approximation of differential equations; application to crystallization processes
Some inverse fractional legendre transforms of gamma function form
Thom Polynomials for Lagrange, Legendre, and Critical Point Function Singularities
On Burkholder function for orthogonal martingales and zeros of Legendre polynomials
A GENERATING FUNCTION OF THE SQUARES OF LEGENDRE POLYNOMIALS
🆕 Influential since 2022
Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and Legendre interval inclusion function
The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications
Efficient Construction of CGL Hash Function Using Legendre Curves
The Associated Operator Legendre Function and the Incomplete Cauchy Problem
Error bounds for Gaussian quadrature formulae with Legendre weight function for analytic integrands
📚 Books worth reading
- 📕Legendre-Tau approximation for functional differential equationsKazufumi Ito · 1984
Solved problems in analysisOrin J. Farrell, Bertram Ross · 1971
Tables of normalized associated Legendre polynomialsS. L. Belousov · 1962- 📕Fonctions sphériques de legendre et fonctions sphéroïdalesLouis Robin · 1957
- 📕Tables of Legendre polynomialsGeorge C. Clark · 1957
- 📕Generalized Associated Legendre Functions and Their ApplicationsNina Virchenko, Iryna Fedotova · 2001
- 📕Legendre polynomialsBritish Association for the Advancement of Science. Committee for the Calculation of Mathematical Tables · 1946
Die Legendre'schen Sätze über die Winkelsumme in DreieckMax Dehn · 1900
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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