In mathematics, Jacobi transform is an

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...

named after the mathematician

Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasio ...

, which uses

Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...

P_n^(x)

as kernels of the transform . The Jacobi transform of a function

F(x)

isDebnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014. :

J\ = f^(n) = \int_^1 (1-x)^\alpha\  (1+x)^\beta \ P_n^(x)\  F(x) \ dx

The inverse Jacobi transform is given by :

J^\ = F(x) = \sum_^\infty \frac f^(n) P_n^(x), \quad \text
 \quad \delta_n =\frac

Some Jacobi transform pairs

References

Integral transforms Mathematical physics {{math-physics-stub