Fast Computation of Singular Oscillatory Fourier Transforms

Table 6

Approximate values in 2-point approximations by the proposed method to the integral ∫¹₋₁(𝑥+1)^−(1/2)(1−𝑥)^−(1/2)𝑒^𝑥𝑒^𝑖𝜔𝑥𝑑𝑥, compared with exact values.

𝜔	500	1000
Exact values	−0.16534892514854+0.03862546212408𝑖	0.12015044233980+0.01747126126322𝑖
Approximate values	−0.16534892514837+0.03862546212408𝑖	0.12015044233979+0.01747126126322𝑖
𝜔	3000	5000
Exact values	−0.03778034068446+0.04543591233617𝑖	−0.03222911938840−0.03366226121525𝑖
Approximate values	−0.03778034068446+0.04543591233617𝑖	−0.03222911938840−0.03366226121525𝑖