라구에르 변환

수학에서 라구에르 변환은 수학자 에드몽 라구에르(Edmond Laguere)의 이름을 딴 적분 변환으로 일반화된 라구에르 다항식 $L{\displaystyle {}_{}^{}}$ ${\displaystyle n_{}^{}}$ ${\displaystyle {}_{}^{}x}$) ${\displaystyle L_{n}^{\alpha }(x)}$를 변환의 커널로 사용한다$$.^[1][2][3][4]

함수 $f{\displaystyle }$( ${\displaystyle x}$) ${\displaystyle f(x)}$의 Laguerre 변환은$$

${\displaystyle L\{f(x)\}={\tilde{f}_{\alpha }(n)=\int_{0}^{0}^{-x}x^{-x}x^{n}^{n}^{n}^{n}^{\alpha }(x)\f(x)\dx}$

역 Laguerre 변환은 다음에 의해 주어진다.

${\displaystyle L^{-1}\{{\tilde {f}}_{\alpha }(n)\}=f(x)=\sum _{n=0}^{\infty }{\binom {n+\alpha }{n}}^{-1}{\frac {1}{\Gamma (\alpha +1)}}{\tilde {f}}_{\alpha }(n)L_{n}^{\alpha }(x)}$

일부 Laguere 변환 쌍

$f(x)\,}$	${\displaystyle {\tilde{f}}_{\\put }(n)\,}$
${\displaystyle x^{a-1}\ a>0\,}$	${\displaystyle {\frac {\Gamma(a+\alpha )\Gamma(n-a+1)}{n!\감마(1-a)}}$
${\displaystyle e^{-ax}\ a>-1\,}$	${\displaystyle {\frac {\Gamma(n+\alpha +1)a^{n}{n!(a+1)^{n+\alpha +1}:{n2}}:$
$\displaystyle \sin agx,\ a>0,\\display =0\,}$	${\displaystyle {\frac{a^{n}{{1+a^{2}^{\frac {n+1}{2}{2}}}}}\sin \왼쪽[n\tan ^{-1}{\frac {1}{a}+\tan ^{1}(-a)\오른쪽]}}$
$\displaystyle \cos 도끼,\ a>0,\ \cHB =0\,}$	${\displaystyle {\frac{a^{n}{{1+a^{2}^{\frac {n+1}{2}{2}}}}}\cos \left[n\tan ^{-1}{\frac {1}{a}+\tan ^{1}(-a)\right]}}$
${\displaystyle L_{m}^{\\알파 }(x)\,}$	${\displaystyle {\binom {n+\alpha }{n}\Gamma(\alpha +1)\delta _{mn}}$
${\displaystyle e^{-ax}L_{m}^{\알파 }(x)\,}$	${\displaystyle {\frac {\Gamma(n+\alpha +1)\Gamma(m+\alpha +1)}{n!m!\Gamma (\alpha +1)}}{\frac {(a-1)^{n-m+\alpha +1}}{a^{n+m+2\alpha +2}}}{}_{2}F_{1}\left(n+\alpha +1;{\frac {m+\alpha +1}{\alpha +1}};{\frac {1}{a^{2}}}\right)}$[5]
${\displaystyle f(x)x^{\reason -\reason }\,}$	${\displaystyle \sum _{m=0}^{n}(m!)^{-1}(\alpha -\beta )_{m}L_{n-m}^{\beta }(x)}$
${\displaystyle e^{x}x^{-\alpha }\Gamma(\alpha ,x)\,}$	${\displaystyle \sum _{n=0}^{\inflat }{\binom {n+\alpha }{n}}{\frac {\gamma(\alpha +1)}{n+1}}}}$
${\displaystyle x^{\put }\\\\\\}$	${\displaystyle \Alpha +\beta +1)\sum _{n=0}^{n}{\binom {n+\alpha }{n}(-\beta )_{n}{n}{\frac {\Alpha +1){\\\\\\\\}감마(n+\알파 +1)}$
${\displaystyle (1-z)^{-(\ju +1)}\exp \leftleft\frac {xz}{z-1}\right),\ z <1,\\\\\\\ \geq 0\,}$	${\displaystyle \sum _{n=0}^{\inful }{\binom {n+\alpha }{n}}\감마(\alpha +1)z^{n}}}}}$
${\displaystyle (xz)^{-\filename /2}e^{z}J_{\alpha }\좌측[2(xz)^{1/2}\우측]\z <1,\ \알파 \geq 0\,}$	${\displaystyle \sum _{n=0}^{\inflt }{\binom {n+\alpha }{n}}{\frac {\Alpha +1)}{\\감마(n+\알파 +1)}z^{n}}$
${\displaystyle {\frac {d}{dx}f(x)\,}$	${\displaystyle {\tilde{f}_{\tile{f}_{k=0}^{n}{\tilde{f}_{\n1}(k)+\{k=0}^{n-1}{\tilde{f}_{\tilde{\}}}}.$
${\displaystyle x{\frac {d}{dx}f(x),\display =0\,}$	${\displaystyle -(n+1){\tilde {f}_{0}(n+1)+n{\tilde {f}_{0}(n)}$
${\displaystyle \int _{0}^{xf(t)dt,\ \propert =0\,}$	${\displaystyle {\tilde{f}_{0}(n)-{\tilde {f}_{0}(n-1)}$
${\displaystyle e^{x}x^{-\frac {d}{dx}}\왼쪽[e^{-x^{-x^{\1}{d}}\f(x)\,}$	${\displaystyle -n{\tilde{f}}_{\display }(n)}$
${\displaystyle \left\{e^}x^{-\x^{d}}{d}}\frac {d}{d}x^{d}}\frac {d}\right\}^{k(x)\,}}$	${\displaystyle(-1)^{k}n^{k}{\tilde {f}_{\display }(n)}$
${\displaystyle L_{n}^{\alpha }(x),\alpha >-1\,}$	${\displaystyle {\frac {\Gamma(n+\alpha +1)}{n!}}}$
${\displaystyle xL_{n}^{\alpha }(x),\alpha >-1\,}$	${\displaystyle {\frac {\Gamma(n+\alpha +1)}{n!}}}(2n+1+\cHB )}$
${\displaystyle {\frac {1}{\pi }}\int _{0}^{\infty }e^{-t}f(t)dt\int _{0}^{\pi }e^{{\sqrt {xt}}\cos \theta }\cos({\sqrt {xt}}\sin \theta )g(x+t-2{\sqrt {xt}}\cos \theta )d\theta ,\alpha =0\,}$	${\displaystyle {\tilde{f}_{0}(n){\tilde {g}_{0}(n)}$
${\displaystyle {\frac {\Gamma (n+\alpha +1)}{{\sqrt {\pi }}\Gamma (n+1)}}\int _{0}^{\infty }e^{-t}t^{\alpha }f(t)dt\int _{0}^{\pi }e^{-{\sqrt {xt}}\cos \theta }\sin ^{2\alpha }\theta g(x+t+2{\sqrt {xt}}\cos \theta ){\frac {J_{\alpha -1/2}({\sqrt {xt}}\sin \theta )}{[({\sqrt {xt}}\sin \theta )/2]^{\alpha -1/2}}}d\theta \,}$	${\displaystyle {\tilde{f}_{\tilde{g}}{\tilde{g}}{\displaysty }$[6]

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