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We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which can be considered as a generalization of the differential equation for the zero-order cylinder functions. Since the asymptotics of our entire part of the Hankel function contains logarithmic terms, this new function can be used to derive asymptotic expansions of some other functions whose asymptotics contain logarithmic terms too. In the Fourier image it can be shown that these expansions are not merely asymptotic but actually convergent.

The Fourier transform of a function related to cylinder functions and asymptotic expansions with logarithmic terms