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^{2}\theta }}}.} This is Legendre's trigonometric form of the elliptic integral; substituting t = sin θ and x = sin φ, one obtains Jacobi's algebraic form: F...

fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. Jacobi was born of Ashkenazi Jewish...

-function The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on...

Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's...

{\displaystyle K(\cdot )} is the complete elliptic integral of the first kind. As discussed above, the Legendre polynomials obey the three-term recurrence...

-js=\mathrm {cd} (w,1/\xi )} where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields: 1 + ϵ 2 c d...

to proceed to calculate the elliptic integral. Given Eq. 3 and the Legendre polynomial solution for the elliptic integral: K ( k ) = π 2 ∑ n = 0 ∞ ( (...

description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents...

Legendre form Nome Quarter period Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi's elliptic functions...

compute elliptic integrals, which are used, for example, in elliptic filter design. The arithmetic–geometric mean is connected to the Jacobi theta function...

also be expressed by the Legendre-Form: These functions can be displayed directly by using the incomplete elliptic integral of the first kind:[citation...

new methods. Examples include: the development of elliptic integrals (Legendre 1811) and elliptic functions (Weierstrass 1861); the development of differential...

work on elliptic functions. Abel's starting point were the elliptic integrals which had been studied in great detail by Adrien-Marie Legendre. He began...

Several orthogonal polynomials, including Jacobi polynomials P(α,β) n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer...

energy. The action S {\displaystyle S} in Hamilton's principle is the Legendre transformation of the action in Maupertuis' principle. The concepts and...

published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument...

Prime-counting function Meissel–Lehmer algorithm Offset logarithmic integral Legendre's constant Skewes' number Bertrand's postulate Proof of Bertrand's...

the theory of Abelian integrals and continued fractions. As written in his last letter, Galois passed from the study of elliptic functions to consideration...

representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician...

letter to Encke. Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order), Sohncke in 1837 (7th and other orders)...