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Legendre transform (integral transform)

From Wikipedia, the free encyclopedia

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function is[1][2][3]

The inverse Legendre transform is given by

Associated Legendre transform

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Associated Legendre transform is defined as

The inverse Legendre transform is given by

Some Legendre transform pairs

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References

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  1. ^ Debnath, Lokenath; Dambaru Bhatta (2007). Integral transforms and their applications (2nd ed.). Boca Raton: Chapman & Hall/CRC. ISBN 9781482223576.
  2. ^ Churchill, R. V. (1954). "The Operational Calculus of Legendre Transforms". Journal of Mathematics and Physics. 33 (1–4): 165–178. doi:10.1002/sapm1954331165. hdl:2027.42/113680.
  3. ^ Churchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100.