In mathematics, Jacobi transform is an

integral transform In mathematics, an integral transform is a type of transform that 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 charac ...

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. Biography Jacobi was ...

, which uses

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

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