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Fonction lemniscatique Funkcje lemniskaty Лемніскатна еліптична функція Lemniscate elliptic functions Función elíptica lemniscática Lemniskatischer Sinus
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Der lemniskatische Sinus oder sinus lemniscatus (kurz sinlemn oder ) ist eine spezielle, von dem Mathematiker Carl Friedrich Gauß eingeführte mathematische Funktion. Der lemniskatische Sinus entspricht derjenigen Funktion für die Lemniskate, die der Sinus für den Kreis ist. Der lemniskatische Cosinus (kurz coslemn oder ) leitet sich direkt von ab. Beides sind die historisch ersten, heute so genannten elliptischen Funktionen. Nach der Definition durch Jacobi ist der Kehrwert der Quadratwurzel aus Zwei der elliptische Modul der lemniskatischen Funktionen. En matemáticas, una función elíptica lemniscática es un tipo de función elíptica relacionada con la longitud del arco de una lemniscata de Bernoulli, como el seno lemniscático (abreviado sinlemn o ) y el coseno lemniscático (abreviado coslemn o ). Estas funciones matemáticas especiales, introducidas por el matemático Carl Friedrich Gauss, se corresponden con las funciones seno coseno de una circunferencia. Históricamente, son las dos primeras de las que posteriormente serían conocidas como funciones elípticas.​ Funkcje lemniskaty – szczególny przykład funkcji eliptycznych, powstającyvh przez odwrócenie całki eliptycznej Całki te pojawiły się po raz pierwszy przy obliczeniu długości łuku lemniskaty Bernoulliego w pracach G. Fagnano z 1715 roku. Funkcje lemniskaty wprowadził Carl Friedrich Gauss w 1797 roku. Są dwie funkcje lemniskaty: gdzie: . In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate functions have periods related to a number 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) 3.141592..., ratio of perimeter to diameter of a circle. У математиці лемніскатна еліптична функція — це еліптична функція, що пов'язана з довжиною дуги лемніскати Бернуллі. Вперше вона була досліджена у 1718 році, та пізніше Леонардом Ейлером, Карлом Фрідріхом Гаусом та іншими. Лемніскатні функції синуса та косинуса, для позначення яких зазвичай використовують символи й (іноді , або та ), є аналогами тригонометричних функцій синуса та косинуса.У той час як тригонометричний синус пов'язує довжину дуги з довжиною хорди в колі одиничного діаметра , лемніскатний синус пов'язує довжину дуги з довжиною хорди лемніскати . En mathématiques, les fonctions lemniscatiques sont des fonctions elliptiques liées à la longueur d'arc d'une lemniscate de Bernoulli ; ces fonctions ont beaucoup d'analogies avec les fonctions trigonométriques. Elles ont été étudiées par Giulio Fagnano en 1718 ; leur analyse approfondie, et en particulier la détermination de leurs périodes, a été obtenue par Carl Friedrich Gauss en 1796. Ces fonctions ont un réseau de périodes carré, et sont étroitement reliées à la fonction elliptique de Weierstrass dont les invariants sont g2 = 1 et g3 = 0. Dans le cas des fonctions lemniscatiques, ces périodes (ω1 et iω1) sont liées à la constante de Gauss G ; on a (où Γ est la fonction gamma).
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A fast algorithm, returning approximations to , is the following: This is effectively using the arithmetic-geometric mean and is based on Landen's transformations.
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En matemáticas, una función elíptica lemniscática es un tipo de función elíptica relacionada con la longitud del arco de una lemniscata de Bernoulli, como el seno lemniscático (abreviado sinlemn o ) y el coseno lemniscático (abreviado coslemn o ). Estas funciones matemáticas especiales, introducidas por el matemático Carl Friedrich Gauss, se corresponden con las funciones seno coseno de una circunferencia. Históricamente, son las dos primeras de las que posteriormente serían conocidas como funciones elípticas.​ In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead) are analogous to the trigonometric functions sine and cosine. While the trigonometric sine relates the arc length to the chord length in a unit-diameter circle the lemniscate sine relates the arc length to the chord length of a lemniscate The lemniscate functions have periods related to a number 2.622057... called the lemniscate constant, the ratio of a lemniscate's perimeter to its diameter. This number is a quartic analog of the (quadratic) 3.141592..., ratio of perimeter to diameter of a circle. As complex functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods and are a special case of two Jacobi elliptic functions on that lattice, . Similarly, the hyperbolic lemniscate sine slh and hyperbolic lemniscate cosine clh have a square period lattice with fundamental periods The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function . Funkcje lemniskaty – szczególny przykład funkcji eliptycznych, powstającyvh przez odwrócenie całki eliptycznej Całki te pojawiły się po raz pierwszy przy obliczeniu długości łuku lemniskaty Bernoulliego w pracach G. Fagnano z 1715 roku. Funkcje lemniskaty wprowadził Carl Friedrich Gauss w 1797 roku. Są dwie funkcje lemniskaty: gdzie: . У математиці лемніскатна еліптична функція — це еліптична функція, що пов'язана з довжиною дуги лемніскати Бернуллі. Вперше вона була досліджена у 1718 році, та пізніше Леонардом Ейлером, Карлом Фрідріхом Гаусом та іншими. Лемніскатні функції синуса та косинуса, для позначення яких зазвичай використовують символи й (іноді , або та ), є аналогами тригонометричних функцій синуса та косинуса.У той час як тригонометричний синус пов'язує довжину дуги з довжиною хорди в колі одиничного діаметра , лемніскатний синус пов'язує довжину дуги з довжиною хорди лемніскати . Періоди лемніскатних функції пов'язані з числом , яке називають лемніскатною константою, що є відношенням лемніскатного периметра до його діаметра. Функції та мають квадратну (кратну гауссовим цілим числам) з , і є окремим випадком двох еліптичних функцій Якобі на цій ґратці, , . Аналогічно, гіперболічні лемніскатичні функції та мають квадратну періодичну ґратку з фундаментальними періодами . Лемніскатні функції та гіперболічні функції пов'язані з еліптичною функцією Веєрштраса . Der lemniskatische Sinus oder sinus lemniscatus (kurz sinlemn oder ) ist eine spezielle, von dem Mathematiker Carl Friedrich Gauß eingeführte mathematische Funktion. Der lemniskatische Sinus entspricht derjenigen Funktion für die Lemniskate, die der Sinus für den Kreis ist. Der lemniskatische Cosinus (kurz coslemn oder ) leitet sich direkt von ab. Beides sind die historisch ersten, heute so genannten elliptischen Funktionen. Nach der Definition durch Jacobi ist der Kehrwert der Quadratwurzel aus Zwei der elliptische Modul der lemniskatischen Funktionen. En mathématiques, les fonctions lemniscatiques sont des fonctions elliptiques liées à la longueur d'arc d'une lemniscate de Bernoulli ; ces fonctions ont beaucoup d'analogies avec les fonctions trigonométriques. Elles ont été étudiées par Giulio Fagnano en 1718 ; leur analyse approfondie, et en particulier la détermination de leurs périodes, a été obtenue par Carl Friedrich Gauss en 1796. Ces fonctions ont un réseau de périodes carré, et sont étroitement reliées à la fonction elliptique de Weierstrass dont les invariants sont g2 = 1 et g3 = 0. Dans le cas des fonctions lemniscatiques, ces périodes (ω1 et iω1) sont liées à la constante de Gauss G ; on a (où Γ est la fonction gamma).
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Gaussian_lemniscate_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Gaussian_lemniscate_sine
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dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Weierstrass_elliptic_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Cl_(elliptic_function)
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Cl_(function)
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Giulio_Carlo_de'_Toschi_di_Fagnano
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Cosine_lemniscate
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dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Cosine_lemniscate_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Cosinus_lemniscatus
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Coslemn
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sl_(function)
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Dixon_elliptic_functions
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sl_(elliptic_function)
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Modular_lambda_function
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Inverse_lemniscatic_elliptic_function
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_case
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_cos
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_cosine
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_cosine_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_elliptic_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_sin
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_sine
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_sine_function
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dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscatic_case
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscatic_cosine
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dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscatic_elliptic_functions
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscatic_sine
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sine_lemniscate
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sine_lemniscate_function
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sinlemn
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Sinus_lemniscatus
dbo:wikiPageWikiLink
dbr:Lemniscate_elliptic_functions
dbo:wikiPageRedirects
dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_function
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
dbr:Lemniscate_functions
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dbr:Lemniscate_elliptic_functions
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dbr:Lemniscate_elliptic_functions
Subject Item
wikipedia-en:Lemniscate_elliptic_functions
foaf:primaryTopic
dbr:Lemniscate_elliptic_functions