Mehler–Fock transform - Alchetron, The Free Social Encyclopedia

In mathematics, the Mehler–Fock transform is an integral transform introduced by Mehler (1881) and rediscovered by Fock (1943).

It is given by

F ( x ) = ∫ 0 ∞ P i t − 1 / 2 ( x ) f ( t ) d t , ( 1 ≤ x ≤ ∞ ) ,

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

f ( t ) = t tanh ⁡ ( π t ) ∫ 1 ∞ P i t − 1 / 2 ( x ) F ( x ) d x , ( 0 ≤ t ≤ ∞ ) .

References

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