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.

我们考虑一个函数，该函数是通过消除其对数奇异性由零阶第一类汉克尔函数获得的。这个函数，除了作为加法项的零阶贝塞尔函数的倍数之外，可以被认为是汉克尔函数的“整个部分”。我们通过考虑一个五阶常微分方程来构造这个函数的傅里叶变换，该方程可以被认为是零阶圆柱函数的微分方程的推广。由于我们整个 Hankel 函数部分的渐近包含对数项，这个新函数可用于推导一些其他函数的渐近展开，这些函数的渐近也包含对数项。在傅里叶图像中可以看出，这些展开不仅是渐近的，而且实际上是收敛的。