Transformada de Legendre (transformación integral)

En matemáticas, Transformación legendaria es una transformación integral llamada después del matemático Adrien-Marie Legendre, que utiliza polinomios Legendre ${displaystyle P_{n}(x)}$ como núcleos de la transformación. Legendre transform es un caso especial de transformación Jacobi.

La leyenda transforma una función ${displaystyle f(x)}$ es

${displaystyle {mathcal {}_{n}{f(x)}={tilde {f}(n)=int _{-1}{1}P_{n}(x) f(x) dx}$

La transformada inversa de Legendre viene dada por

${fnK} {fn} {fn} {f} {f} {f}=f(x)=sum _{n=0}{infty }{frac {2n+1}{f} {f} {c} {cH00} {c} {cH00} {cH00}}} {cH00}}}}}}}}} {f}}}}} {f} {f}}}} {f}}}} {f}}}}} {f}}}} {f}}}} {f}}}}}}}}}} {f}}}}} {f} {f}}} {f}}}} {f}}}} {f} {f} {f}}} {f}}}}} {f} {f}}}}} {f}}}}}}}}}}}}}}} {$

Transformación de Legendre asociada

La transformación de Legendre asociada se define como

${fnMicrosoft Sans Serif} {f}= {f} {f} {f}(n,m)=int _{1}{1}(1-x^{2})^{-m/2}P_{n}{m}(x) f(x) dx}$

La transformada inversa de Legendre viene dada por

${f} {f} {f} {f} {f} {f} {f} {f} {f}f(x)=sum _{n=0} {f} {m} {m} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m} {m} {m} {m} {m}} {m} {m} {m}}} {m} {m} {m}}} {m}}}}}} {m} {m} {m}} {m} {m} {m} {m} {m} {m} {m} {m} {m}} {m}}}}} {m} {m} {m}} {m}}} {m}}} {m} {m}$

Algunos pares de transformaciones de Legendre

${displaystyle f(x),}$	${displaystyle {f}(n),}$
${displaystyle x^{n},}$	${displaystyle {frac {2^{n+1}(n)}{2}{(2n+1)}}}}$
${displaystyle e^{ax},}$	${fnMicroc} {2ccH00} Yo...$
${displaystyle e^{iax},}$	${fnfnMicroc {2fn\fn\fn\fn\fn\\fn\fn\\\fn\\\\\fn\\\\\\\\fn\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\\\\\\\fn\\\\\\\\\\\\\\\\\ - ¿Qué? }$
${displaystyle xf(x),}$	${displaystyle {frac {1}{2n+1} {n+1){tilde {f}(n+1)+n{tilde {f}(n-1)}}$
${displaystyle (1-x^{2}} {-1/2},}$	${displaystyle pi P_{n} {2}(0)}$
${displaystyle [2(a-x)]$	${displaystyle Q_{n}(a)}$
${displaystyle (1-2ax+a^{2})^{-1/2}, Накованиваниваниванить1,}$	${displaystyle 2a^{n}(2n+1)}{-1}$
${displaystyle (1-2ax+a^{2})^{-3/2}, Накованиваниваниванить1,}$	${displaystyle 2a^{n}(1-a^{2}{-1}$
${displaystyle int _{0}{a}{frac {t^{b-1},dt}{(1-2xt+t^{2}}}}}}$	${displaystyle {frac {2a^{n+b}{(2n+1)(n+b)}}}$
${displaystyle {frac {dx}left[(1-x^{2}{frac {d}{dx}right]f(x),}$	${displaystyle -n(n+1){tilde {f}(n)}$
${displaystyle left{frac {dx}left[(1-x^{2}{frac {d}derecha]derecha}{k}f(x),}$	${fnMicrosoft Sans Serif} {f} {f} {f} {f} {f} {f} {f} {f}}}} {f}}}} {f}}}} {f}} {f}} {f}}} {f}}}}}}}}}}$
${displaystyle {frac {f(x)}{4}-{frac {dx}left[(1-x^{2}){frac {dx}right]f(x),}$	${displaystyle left(n+{2}right)}{2}{2}{tilde {f}(n)}$
${displaystyle ln(1-x),}$	${displaystyle {begin{cases}2(ln 2-1), sensiblen=0\-{frac {2}{n(n+1)}}}} {fnfn1}},}}$
${displaystyle f(x)*g(x),}$	${displaystyle {tilde {f}(n){tilde {g}(n)}$
${displaystyle int _{-1}{x}f(t),dt,}$	${fnMicrosoft Sans Serif} {f} {f} {f} {f} {f}(1), limitada=0\\\\{tilde {f}(n-1)-{tilde {f} {n+1)}{2n+1}}}}}}} {f}}}}}}$
${displaystyle {frac {dx}g(x), g(x)=int _{-1}^{x}f(t),dt}$	${displaystyle g(1)-int _{-1}{1}g(x){frac {d}}P_{n}(x),dx}$