Mehler–Fock transform

In mathematics, the Mehler–Fock transform is an integral transform introduced by Template:Harvs and rediscovered by Template:Harvs.

It is given by

$F(x)={\int }_0^{\mathrm{\infty }}{P}_{it-1/2}(x)f(t)dt,(1\le x\le \mathrm{\infty }),$

where P is a Legendre function of the first kind.

Under appropriate conditions, the following inversion formula holds:

$f(t)=t\tanh (\pi t){\int }_1^{\mathrm{\infty }}{P}_{it-1/2}(x)F(x)dx,(0\le t\le \mathrm{\infty }).$

References

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