Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals

Table 3

Rules	Integral
	Approximate value of 𝐼3=𝑃∫^(−1+𝑖)/4(1+𝑖)/4(tan^−1𝑧/𝑧)𝑑𝑧	Absolute error
𝑅1(𝑓)	−0.506612+0.492763∗𝑖	2.2×10^−6
𝑅2(𝑓)	−0.506614+0.492765∗𝑖	1.0×10^−6
𝑅3(𝑓)	−0.506614+0.492765∗𝑖	1.0×10^−6
Exact value	−0.506614+0.492764∗𝑖	**