Jacobi transformation

In the mathematics which is Jacobi transform a after the mathematician Carl Gustav Jacob Jacobi named integral transformation , the Jacobi polynomials used as cores of transformation. ${\displaystyle P_{n}^{\alpha ,\beta }(x)}$

The Jacobian transformation of a function is ${\displaystyle F}$

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

The inverse Jacobi transform is given by

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

Some Jacobi transform pairs

${\displaystyle F(x)\,}$	${\displaystyle f^{\alpha ,\beta }(n)\,}$
${\displaystyle x^{m},\ m<n\,}$	${\displaystyle 0}$
${\displaystyle x^{n}\,}$	${\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}}$
${\displaystyle P_{m}^{\alpha ,\beta }(x)\,}$	${\displaystyle \delta _{mn}}$
${\displaystyle (1+x)^{a-\beta }\,}$	${\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}}$
${\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1\,}$	${\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}}$
${\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1\,}$	${\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}}$
${\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1\,}$	${\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}}$
${\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}$	${\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)}$
${\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}$	${\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)}$

Web links

• Lexicon of Mathematics: Jacobi Transformation, In: Spektrum.de
• Jacobi Transformation of a Symmetric Matrix, Author: Christina C. Lee, In: albi3ro (English)
• Jacobi matrix of transformation, p. 8, In: Mathematik.Uni.Stuttgart (PDF)
• Jacobi Transformation, In: Mathworld.Wolfram
• Prof. D. Salomon Analysis II, Math, Phy, Chab, Jacobi-Transformation, sample calculation pp. 1-9, In: Math.ethz.ch
• Jacobi transformation, non-parametric statistical methods by Herbert Büning, Götz Trenkler, Walter de Gruyter, Berlin - New York 1994, ISBN 3-11-013860-3 in the Google book search
• Jacobi-Transformation, Lexikon der Mathematik: Volume 3: Inp bis Mon, by Guido Walz (Ed.), Springer Spectrum, ISBN 978-3-662-53501-1, Heidelberg 2001 in the Google book search
• Jacobi-Transformation, CRC Concise Encyclopedia of Mathematics by Eric W. Weisstein, 2003 by Chapman & Hall / CRC, ISBN 1-58488-347-2 in the Google book search
• Jacobi transformation, discontinuous GALERKIN processes in space and time for simulating transport processes by Sandra Carstens, dissertation 2012, Kassel University, ISBN 978-3-86219-434-6 in the Google book search
• Jacobi Transformation, Applied Finite Element Analysis by G. Ramamurty, IK International Publishing House Pvt. Lt, New Delhi - Bangalore 2010 in google book search

Individual evidence

1. ↑ Debnath, L. "On Jacobi transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.
2. ↑ Debnath, L. "Solution of partial differential equations by Jacobi transform." Bulletin of the Calcutta Mathematical Society 59.3-4 (1967): 155.
3. ^ Scott, EJ "Jacobi transforms." (1953).
4. ↑ Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.