Mehler–Fock transform

In mathematics, the Mehler–Fock transform is an integral transform introduced by {{harvs|txt|authorlink=Gustav Ferdinand Mehler|last=Mehler|year=1881}} and rediscovered by {{harvs|txt|authorlink=Vladimir Fock|last=Fock|year=1943}}.

It is given by

: $F(x) =\int_0^\infty P_{it-1/2}(x)f(t) dt,\quad (1 \leq x \leq \infty),$

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

: $f(t) = t \tanh(\pi t) \int_1^\infty P_{it-1/2}(x)F(x) dx ,\quad (0 \leq t \leq \infty).$

References

{{eom|id=m/m063340|first=Yu.A.|last= Brychkov|first2=A.P. |last2=Prudnikov}}
{{Citation | last1=Fock | first1=V. A. | title=On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index | mr=0009665 | year=1943 | journal=C. R. (Doklady) Acad. Sci. URSS |series=New Series | volume=39 | pages=253–256}}
{{Citation | last1=Mehler | first1=F. G. | title=Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung | publisher=Springer Berlin / Heidelberg | language=German | doi=10.1007/BF01445847 | year=1881 | journal=Mathematische Annalen | issn=0025-5831 | volume=18 | issue=2 | pages=161–194}}
{{eom|id=m/m120190|first=S. B. |last=Yakubovich}}

Category:Integral transforms