Abstract
The Hankel transform \(\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x\) is studied for integer \(n\geqslant -1\) and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f(x) and its derivatives are imposed. It is necessary to underline that similar formulas obtained previously are in fact asymptotic expansions only valid when q tends to infinity. If one of the conditions is violated, our series become asymptotic series. The validity of the formulas is illustrated by a number of examples.
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Notes
The extension of Ramanujan’s master theorem for \(A \in (0,\pi ]\) was done in [23].
All even-order derivatives are positive, while all odd-order derivatives are negative, or vice versa.
For a special case \(a=c\), this sum is given by formula [27, 4.2.7.37].
In a sum of three terms, the poles cancel each other.
It corresponds to an alternative definition of the Hankel transform: \(\bar{\mathcal {H}}_\nu [f(x)](q) = \int _0^{\infty } \!\! f(x) J_\nu (q x) \,\mathrm{d}x\).
We used the fact that \(xf(x)J_1(q x)|_{x=0} = xf(x)J_1(q x)|_{x=\infty } =0\).
Taking into account that \(xf(x)J_\nu (q x)|_{x=0} = xf(x)J_\nu (q x)|_{x=\infty } =0\).
The same is true for \(\Phi (t) = J_0(t)\), \(f(t) = t\cos t, t\sin t, tJ_\nu (t)\), etc.
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Appendix A
Appendix A
Here we calculate a series which defines integral (28). After using the relation
After putting \(m=n+k\), we find
Using relation
we obtain (\(q > a+c\))
where
is the hypergeometric series of two variables [24]. It is the uniformly and absolutely convergence series for \(\sqrt{|x|} + \sqrt{|y|} < 1\).
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Kisselev, A.V. Exact expansions of Hankel transforms and related integrals. Ramanujan J 55, 349–367 (2021). https://doi.org/10.1007/s11139-020-00274-x
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DOI: https://doi.org/10.1007/s11139-020-00274-x