Mehler–Fock transform
Open-knowledge reference entry
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
≤
∞
)
,
{\displaystyle F(x)=\int _{0}^{\infty }P_{it-1/2}(x)f(t)dt,\quad (1\leq x\leq \infty ),}
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
≤
∞
)
.
{\displaystyle f(t)=t\tanh(\pi t)\int _{1}^{\infty }P_{it-1/2}(x)F(x)dx,\quad (0\leq t\leq \infty ).}
References
Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Mehler–Fock transform", Encyclopedia of Mathematics, EMS Press
Fock, V. A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. (Doklady) Acad. Sci.
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