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Legendre transform (integral transform)

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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

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Associated Legendre transform

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Associated Legendre transform is defined as

The inverse Legendre transform is given by

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Some Legendre transform pairs

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References

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