Mehler-Fock transform

Mehler–Fok transform

The integral transform

(1)

where is the Legendre function of the first kind (cf. Legendre functions). If , the function is locally integrable on and , then the following inversion formula is valid:

(2)

The Parseval identity. Consider the Mehler–Fock transform and its inverse defined by the equalities

If , , are arbitrary real-valued functions satisfying the conditions

then

The generalized Mehler–Fock transform and the corresponding inversion formula are:

(3)

and

(4)

where are the associated Legendre functions of the first kind. For formulas (3) and (4) reduce to (1) and (2); for , , formulas (3) and (4) lead to the Fourier cosine transform, and for , to the Fourier sine transform. The transforms (1) and (2) were introduced by F.G. Mehler [1]. The basic theorems were proved by V.A. Fock [V.A. Fok].

References

[1]	F.G. Mehler, "Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Electricitätsvertheilung" Math. Ann. , 18 (1881) pp. 161–194
[2]	V.A. Fok, "On the representation of an arbitrary function by an integral involving Legendre functions with complex index" Dokl. Akad. Nauk SSSR , 39 (1943) pp. 253–256 (In Russian)
[3]	V.A. Ditkin, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–75 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82

References

[a1]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)