Lagrange reversion theorem (original) (raw)
In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that Then for any function g, for small enough y: If g is the identity, this becomes In which case the equation can be derived using perturbation theory. Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation.
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| dbo:abstract | In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that Then for any function g, for small enough y: If g is the identity, this becomes In which case the equation can be derived using perturbation theory. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration. Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation. (en) En matemáticas, el teorema de la reversión de Lagrange nos da la expansión en serie de potencias o en serie formal de potencias de ciertas funciones implícitamente definidas, de hecho, de composiciones de tales funciones. Sea una función de e definida a partir de otra función tal que Entonces, cualquier función se puede desarrollar en serie de Taylor alrededor de para pequeño, es decir, se tiene Si es la función identidad, es decir, , En 1770, Joseph Louis Lagrange (1736–1813) publicó su solución en serie de potencias de la ecuación implícita de antes mencionada. Sin embargo, su solución era algo engorrosa, pues utilizó desarrollos en serie de logaritmos. En 1780, Pierre-Simon Laplace (1749–1827) publicó una prueba más simple del teorema, la cual estaba basada en relaciones entre derivadas parcialescon respecto a la variable y al parámetro . Charles Hermite (1822–1901) presentó la prueba más sencilla del teorema usando integración de contorno. El teorema de reversión de Lagrange se usa para obtener solucciones numéricas de la ecuación de Kepler. (es) |
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| rdfs:comment | In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that Then for any function g, for small enough y: If g is the identity, this becomes In which case the equation can be derived using perturbation theory. Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation. (en) En matemáticas, el teorema de la reversión de Lagrange nos da la expansión en serie de potencias o en serie formal de potencias de ciertas funciones implícitamente definidas, de hecho, de composiciones de tales funciones. Sea una función de e definida a partir de otra función tal que Entonces, cualquier función se puede desarrollar en serie de Taylor alrededor de para pequeño, es decir, se tiene Si es la función identidad, es decir, , El teorema de reversión de Lagrange se usa para obtener solucciones numéricas de la ecuación de Kepler. (es) |
| rdfs:label | Teorema de reversión de Lagrange (es) Lagrange reversion theorem (en) |
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