Transformada de Laguerre From Wikipedia, the free encyclopedia En matemáticas, la transformada de Laguerre es una transformada integral que lleva el nombre del matemático Edmond Laguerre. Utiliza los polinomios de Laguerre L n α ( x ) {\displaystyle L_{n}^{\alpha }(x)} generalizados como núcleos de la transformada.[1][2][3][4] La transformada de Laguerre de una función f ( x ) {\displaystyle f(x)} es: L { f ( x ) } = f ~ α ( n ) = ∫ 0 ∞ e − x x α L n α ( x ) f ( x ) d x {\displaystyle L\{f(x)\}={\tilde {f}}_{\alpha }(n)=\int _{0}^{\infty }e^{-x}x^{\alpha }\ L_{n}^{\alpha }(x)\ f(x)\ dx} La transformada inversa de Laguerre viene dada por: L − 1 { f ~ α ( n ) } = f ( x ) = ∑ n = 0 ∞ ( n + α n ) − 1 1 Γ ( α + 1 ) f ~ α ( n ) L n α ( x ) {\displaystyle L^{-1}\{{\tilde {f}}_{\alpha }(n)\}=f(x)=\sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}^{-1}{\frac {1}{\Gamma (\alpha +1)}}{\tilde {f}}_{\alpha }(n)L_{n}^{\alpha }(x)} Algunos pares de transformadas de Laguerre Más información , ... f ( x ) {\displaystyle f(x)\,} f ~ α ( n ) {\displaystyle {\tilde {f}}_{\alpha }(n)\,} x a − 1 , a > 0 {\displaystyle x^{a-1},\ a>0\,} Γ ( a + α ) Γ ( n − a + 1 ) n ! Γ ( 1 − a ) {\displaystyle {\frac {\Gamma (a+\alpha )\Gamma (n-a+1)}{n!\Gamma (1-a)}}} e − a x , a > − 1 {\displaystyle e^{-ax},\ a>-1\,} Γ ( n + α + 1 ) a n n ! ( a + 1 ) n + α + 1 {\displaystyle {\frac {\Gamma (n+\alpha +1)a^{n}}{n!(a+1)^{n+\alpha +1}}}} sin a x , a > 0 , α = 0 {\displaystyle \sin ax,\ a>0,\ \alpha =0\,} a n ( 1 + a 2 ) n + 1 2 sin [ n tan − 1 1 a + tan − 1 ( − a ) ] {\displaystyle {\frac {a^{n}}{(1+a^{2})^{\frac {n+1}{2}}}}\sin \left[n\tan ^{-1}{\frac {1}{a}}+\tan ^{-1}(-a)\right]} cos a x , a > 0 , α = 0 {\displaystyle \cos ax,\ a>0,\ \alpha =0\,} a n ( 1 + a 2 ) n + 1 2 cos [ n tan − 1 1 a + tan − 1 ( − a ) ] {\displaystyle {\frac {a^{n}}{(1+a^{2})^{\frac {n+1}{2}}}}\cos \left[n\tan ^{-1}{\frac {1}{a}}+\tan ^{-1}(-a)\right]} L m α ( x ) {\displaystyle L_{m}^{\alpha }(x)\,} ( n + α n ) Γ ( α + 1 ) δ m n {\displaystyle {\binom {n+\alpha }{n}}\Gamma (\alpha +1)\delta _{mn}} e − a x L m α ( x ) {\displaystyle e^{-ax}L_{m}^{\alpha }(x)\,} Γ ( n + α + 1 ) Γ ( m + α + 1 ) n ! m ! Γ ( α + 1 ) ( a − 1 ) n − m + α + 1 a n + m + 2 α + 2 2 F 1 ( n + α + 1 ; m + α + 1 α + 1 ; 1 a 2 ) {\displaystyle {\frac {\Gamma (n+\alpha +1)\Gamma (m+\alpha +1)}{n!m!\Gamma (\alpha +1)}}{\frac {(a-1)^{n-m+\alpha +1}}{a^{n+m+2\alpha +2}}}{}_{2}F_{1}\left(n+\alpha +1;{\frac {m+\alpha +1}{\alpha +1}};{\frac {1}{a^{2}}}\right)} [5] f ( x ) x β − α {\displaystyle f(x)x^{\beta -\alpha }\,} ∑ m = 0 n ( m ! ) − 1 ( α − β ) m L n − m β ( x ) {\displaystyle \sum _{m=0}^{n}(m!)^{-1}(\alpha -\beta )_{m}L_{n-m}^{\beta }(x)} e x x − α Γ ( α , x ) {\displaystyle e^{x}x^{-\alpha }\Gamma (\alpha ,x)\,} ∑ n = 0 ∞ ( n + α n ) Γ ( α + 1 ) n + 1 {\displaystyle \sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}{\frac {\Gamma (\alpha +1)}{n+1}}} x β , β > 0 {\displaystyle x^{\beta },\ \beta >0\,} Γ ( α + β + 1 ) ∑ n = 0 ∞ ( n + α n ) ( − β ) n Γ ( α + 1 ) Γ ( n + α + 1 ) {\displaystyle \Gamma (\alpha +\beta +1)\sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}(-\beta )_{n}{\frac {\Gamma (\alpha +1)}{\Gamma (n+\alpha +1)}}} ( 1 − z ) − ( α + 1 ) exp ( x z z − 1 ) , ‖ z | < 1 , α ≥ 0 {\displaystyle (1-z)^{-(\alpha +1)}\exp \left({\frac {xz}{z-1}}\right),\|z|<1,\ \alpha \geq 0\,} ∑ n = 0 ∞ ( n + α n ) Γ ( α + 1 ) z n {\displaystyle \sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}\Gamma (\alpha +1)z^{n}} ( x z ) − α / 2 e z J α [ 2 ( x z ) 1 / 2 ] , ‖ z | < 1 , α ≥ 0 {\displaystyle (xz)^{-\alpha /2}e^{z}J_{\alpha }\left[2(xz)^{1/2}\right],\|z|<1,\ \alpha \geq 0\,} ∑ n = 0 ∞ ( n + α n ) Γ ( α + 1 ) Γ ( n + α + 1 ) z n {\displaystyle \sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}{\frac {\Gamma (\alpha +1)}{\Gamma (n+\alpha +1)}}z^{n}} d d x f ( x ) {\displaystyle {\frac {d}{dx}}f(x)\,} f ~ α ( n ) − α ∑ k = 0 n f ~ α − 1 ( k ) + ∑ k = 0 n − 1 f ~ α ( k ) {\displaystyle {\tilde {f}}_{\alpha }(n)-\alpha \sum _{k=0}^{n}{\tilde {f}}_{\alpha -1}(k)+\sum _{k=0}^{n-1}{\tilde {f}}_{\alpha }(k)} x d d x f ( x ) , α = 0 {\displaystyle x{\frac {d}{dx}}f(x),\alpha =0\,} − ( n + 1 ) f ~ 0 ( n + 1 ) + n f ~ 0 ( n ) {\displaystyle -(n+1){\tilde {f}}_{0}(n+1)+n{\tilde {f}}_{0}(n)} ∫ 0 x f ( t ) d t , α = 0 {\displaystyle \int _{0}^{x}f(t)dt,\ \alpha =0\,} f ~ 0 ( n ) − f ~ 0 ( n − 1 ) {\displaystyle {\tilde {f}}_{0}(n)-{\tilde {f}}_{0}(n-1)} e x x − α d d x [ e − x x α + 1 d d x ] f ( x ) {\displaystyle e^{x}x^{-\alpha }{\frac {d}{dx}}\left[e^{-x}x^{\alpha +1}{\frac {d}{dx}}\right]f(x)\,} − n f ~ α ( n ) {\displaystyle -n{\tilde {f}}_{\alpha }(n)} { e x x − α d d x [ e − x x α + 1 d d x ] } k f ( x ) {\displaystyle \left\{e^{x}x^{-\alpha }{\frac {d}{dx}}\left[e^{-x}x^{\alpha +1}{\frac {d}{dx}}\right]\right\}^{k}f(x)\,} ( − 1 ) k n k f ~ α ( n ) {\displaystyle (-1)^{k}n^{k}{\tilde {f}}_{\alpha }(n)} L n α ( x ) , α > − 1 {\displaystyle L_{n}^{\alpha }(x),\alpha >-1\,} Γ ( n + α + 1 ) n ! {\displaystyle {\frac {\Gamma (n+\alpha +1)}{n!}}} x L n α ( x ) , α > − 1 {\displaystyle xL_{n}^{\alpha }(x),\alpha >-1\,} Γ ( n + α + 1 ) n ! ( 2 n + 1 + α ) {\displaystyle {\frac {\Gamma (n+\alpha +1)}{n!}}(2n+1+\alpha )} 1 π ∫ 0 ∞ e − t f ( t ) d t ∫ 0 π e x t cos θ cos ( x t sin θ ) g ( x + t − 2 x t cos θ ) d θ , α = 0 {\displaystyle {\frac {1}{\pi }}\int _{0}^{\infty }e^{-t}f(t)dt\int _{0}^{\pi }e^{{\sqrt {xt}}\cos \theta }\cos({\sqrt {xt}}\sin \theta )g(x+t-2{\sqrt {xt}}\cos \theta )d\theta ,\alpha =0\,} f ~ 0 ( n ) g ~ 0 ( n ) {\displaystyle {\tilde {f}}_{0}(n){\tilde {g}}_{0}(n)} Γ ( n + α + 1 ) π Γ ( n + 1 ) ∫ 0 ∞ e − t t α f ( t ) d t ∫ 0 π e − x t cos θ sin 2 α θ g ( x + t + 2 x t cos θ ) J α − 1 / 2 ( x t sin θ ) [ ( x t sin θ ) / 2 ] α − 1 / 2 d θ {\displaystyle {\frac {\Gamma (n+\alpha +1)}{{\sqrt {\pi }}\Gamma (n+1)}}\int _{0}^{\infty }e^{-t}t^{\alpha }f(t)dt\int _{0}^{\pi }e^{-{\sqrt {xt}}\cos \theta }\sin ^{2\alpha }\theta g(x+t+2{\sqrt {xt}}\cos \theta ){\frac {J_{\alpha -1/2}({\sqrt {xt}}\sin \theta )}{[({\sqrt {xt}}\sin \theta )/2]^{\alpha -1/2}}}d\theta \,} f ~ α ( n ) g ~ α ( n ) {\displaystyle {\tilde {f}}_{\alpha }(n){\tilde {g}}_{\alpha }(n)} [6] Cerrar Referencias [1]Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014. [2]Debnath, L. "On Laguerre transform." Bull. Calcutta Math. Soc 52 (1960): 69-77. [3]Debnath, L. "Application of Laguerre Transform on heat conduction problem." Annali dell’Università di Ferrara 10.1 (1961): 17-19. [4]McCully, Joseph. "The Laguerre transform." SIAM Review 2.3 (1960): 185-191. [5]Howell, W. T. "CI. A definite integral for legendre functions." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 25.172 (1938): 1113-1115. [6]Debnath, L. "On Faltung theorem of Laguerre transform." Studia Univ. Babes-Bolyai, Ser. Phys 2 (1969): 41-45. Datos: Q30694289 Related Articles
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