Fast Computation of Singular Oscillatory Fourier Transforms

Table 2

(a) Approximate values in 2-point approximations by the proposed method to the integral ∫¹₀𝑥^−(1/2)(𝑥−1)^−(1/2)cos𝑥𝑒^𝑖𝜔𝑥𝑑𝑥, compared with exact values. (b) Approximate values in 2-point approximations by the Clenshaw-Curtis-Filon-type method [31] to the integral ∫¹₀𝑥^−(1/2)(𝑥−1)^−(1/2)cos𝑥𝑒^𝑖𝜔𝑥𝑑𝑥, compared with exact values.

(a)  

𝜔	1000	2000
Exact values	0.06937869322675+0.04527657820424𝑖	0.03655011452354+0.04767073962282𝑖
Approximate values	0.06937869322672+0.04527657820423𝑖	0.03655011452354+0.04767073962281𝑖
𝜔	3000	5000
Exact values	0.01353266279337+0.03766001599991𝑖	0.00974123171200+0.00678449969265𝑖
Approximate values	0.01353266279337+0.03766001599991𝑖	0.00974123171200+0.00678449969265𝑖

(b)  

𝜔	1000	2000
Exact values	0.06937869322675+0.04527657820424𝑖	0.03655011452354+0.04767073962282𝑖
Approximate values	0.06937879634322+0.04527685643561𝑖	0.03655017563845+0.0476703845971𝑖
𝜔	3000	5000
Exact values	0.01353266279337+0.03766001599991𝑖	0.00974123171200+0.00678449969265𝑖
Approximate values	0.01353266884523+0.03766001457824𝑖	0.00974123131452+0.00678449989714𝑖