Efficient numerical methods for the algebraic singular and hypersingular oscillatory Fourier transform with a special oscillator

A recent work (Xu and Liu (2025) [51] ) focused on the efficient computation for the simple case (i.e. k = 1 ) of the highly oscillatory Fourier transform ∫ 0 + ∞ x α f ( x ) ( x − a ) v e i ω x k d x with an algebraic singularity and a hypersingularity, where − 1 < α ≤ 0 , a > 0 , v is an arbitrary positive integer. This paper devises and analyzes two different methods to efficiently calculate this Fourier transform, where 2 k is an arbitrary positive integer. In such case, the problem becomes more challenging. To facilitate and implement these algorithms, we first derive two explicit formulae of the required moments via special functions, such as Meijer G-function. The first method involves a special transformation of the integral, and we partition it into two parts. The first part is further divided into two integrals, which can be efficiently calculated by combining the Clenshaw-Curtis-type method with the complex integration method, respectively. And then the second part can be accurately computed by applying the derived explicit formulae. The second method needs to construct new integration paths and combine the two complex integration methods. In particular, we carry out error estimates of the presented quadrature methods. In the end, by numerical examples, we verify error estimates and demonstrate the precision and efficiency of these methods.