In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form H+(f(t)eiωt)(x)=−int0∞​eiωtt−xf(t)​dt,ω>0,x≥0, where the bar indicates the Cauchy principal value and f is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When x=0, the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of ω are derived for
each fixed x≥0, which clarify the large ω behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of x, we
classify our discussion into three regimes, namely, x=O(1) or
x≫1, 0<x≪1 and x=0. Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency ω increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table