QUADRATIC CONVERGENCE OF THE TANH-SINH ...

QUADRATIC CONVERGENCE OF THE TANH-SINH QUADRATURE RULE 11Therefore,∫1Ê h,(w) ̂K(z, w) = ̂K(z, w)Φ(w) dw2πi ∂S α= 1 ∫π 2 cosh(z) cosh(w)Φ(w)2πi ∂S α4 cosh ( π2 sinh(z) − π 2 sinh(w)) cosh ( π2 sinh(z)) cosh ( πdw,2sinh(w))where(4.13)Φ(z) := Φ(z; h) = Ψ(z) − π cot πz{h−2πi1−exp(−2πiz/h)=, Im(z)> 0, Im(z)< 0.2πi1−exp(2πiz/h)Here Ψ(x) is defined as in (2.4). If we let α → ∞ while keeping ∂S α away from thepoles,(2.5) gives=∫ ∞−∞12πi∫f(x)dx∂S αf(z)Ψ(z) dz −+ π cosh(z)sinh(π sinh(z))∞∑n=−∞π cosh(z)sinh(π sinh(z))∞∑n=−∞Ψ(i arcsin(2n + 1))Ψ (arcsinh (sinh(z) + (2n + 1)i))and equation (2.6) gives=h∞∑n=−∞12πi∫f(nh)∂S αf(z)π cot πzh dz −+ π cosh(z)sinh(π sinh(z))∞∑n=−∞π cosh(z)∞∑( )π · i arcsin(2n + 1)π cotsinh(π sinh(z))hn=−∞( )π · arcsinh(sinh(z) + (2n + 1)i)π cothWe thus have=Ê h,(w) ̂K(z, w)∫1π cosh(z)̂K(z, w)Φ(w) dw −2πi ∂S αsinh(π sinh(z))+ π cosh(z)sinh(π sinh(z))∞∑n=−∞∞∑n=−∞Φ(arcsinh(sinh(z) + (2n + 1)i))Φ(i arcsin(2n + 1))