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.

Hankel变换\（\数学{H} _n [f（x）]（q）= \ int _0 ^ {\ infty} \！\！\，xf（x）J_n（qx）\ mathrm {d} x \ ）研究整数\（n \ geqslant -1 \）和正参数q。证明了在给定函数f（x）及其导数具有特殊条件的情况下，汉克尔变换是由q的倒数的一致且绝对收敛的级数给出的。需要强调的是，先前获得的相似公式实际上仅在q趋于无穷大时才是渐近展开。如果条件之一被违反，我们的级数将成为渐近级数。公式的有效性由许多示例说明。