자코비 변형

수학에서 자코비 변환은 수학자 칼 구스타프 제이콥 자코비의 이름을 딴 적분 변환으로, 자코비 다항식 $P{\displaystyle {}_{}^{}}$ ${\displaystyle {}_{}^{}}$ ,${\displaystyle {\beta }_{}^{}}$(${\displaystyle x}$ ${\displaystyle )}$ ${\displaystyle P_{n}^{$n}^{\$alpha ,\beta }(x)}$을 변환의 커널로$$ 사용한다.^[1][2][3][4]

함수 $F{\displaystyle (x}$) ${\displaystyle F(x)}$의 Jacobi 변환은$$^[5]

${\displaystyle J\{F(x)\}=f^{\alpha ,\beta }=\int _{-1}^{1}{-x)^{\alpha }\{n}^{n}^{n}^{n(x)\f(x)\x\}$

역 자코비 변환은 다음과 같다.

${\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\감마(n+\알파 +\베타 +1)}}$

일부 자코비 변환 쌍

$F(x)\,}$	${\displaystyle f^{\put,\put }(n)\,}$
${\displaystyle x^{m}\ m<n\,}$	${\displaystyle 0}$
${\displaystyle x^{n}\,}$	${\displaystyle n!(\displaystyle +\n+1)\properties _{n}$
${\displaystyle P_{m}^{\\alpha,\beta }(x)\,}$	${\displaystyle \cHB_{n}\cHB_{m,n}}$
${\displaystyle (1+x)^{a-\buffer }\,}$	${\displaystyle {\binom {n+\alpha }{n2}}^{\alpha +a+1}{\frac {\gamma (a+1)\감마(\alpha +1)\감마(a-\beta +1){\\\감마(\알파 +a+n+2)\감마(a-\베타 +n+1)}}}$
${\displaystyle (1-x)^{\sigma -\alpha }\re \sigma >-1\,}$	${\displaystyle {\frac {2^{\buffma +\buffer +1}{n!\감마(\alpha -\sigma )}{\frac {\감마(\sigma +1)\감마(n+\beta +1)\감마(\alpha -\sigma +n)}{\\\}감마(\베타 +\시그마 +n+2)}}}$
${\displaystyle (1-x)^{\sigma -}P_{m}^{\알파,\sigma }(x),\\re \sigma >-1\,}$	${\displaystyle {\frac {2^{\buffer +\ma +1}{m!(n-m)!}}{\frac {\감마(n+\알파 +1)\감마(\알파 +\베타 +m+n+1)\감마(\sigma +m+1)\감마(\알파 -\베타 +1)}{\감마(\알파 +\베타 +n+1)\감마(\알파 +\sigma +m+n+2)\감마(\알파 -\베타 +m+1)}}}$
${\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\nalpha }}^{-\beta },\Q=(1-2xz+z^{2})^{1/2},\z1\,}$	${\displaystyle \sum _{n=0}^{\nft }\ft _{n}z^{n}}}}$
${\displaystyle (1-x)^{-\alpha }^{-\beta }{d}}{dx}\{dx}}\frac {d}}{dx}왼쪽[(1-x)^{\falpha +1}^{d}{dx}\right]F(x)\,},},}$	${\displaystyle -n(n+\reason +\reason +1)f^{\reason,\reason }(n)}$
${\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}$	${\displaystyle(-1)^{k}n^{k}(n+\reason +\1)^{k}f^{\reason,\reason }(n)}$

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