QUADRATIC CONVERGENCE OF THE TANH-SINH ...

QUADRATIC CONVERGENCE OF THE TANH-SINH QUADRATURE RULE 5Lemma 2.4. Suppose, in Lemma 2.3, that f has poles z i with residues Res(z i )inside S α . Then∫ ∞∫1f(x)dx =f(z)Ψ(z) dz + πi ∑ Res(z + i−∞2πi) − πi ∑ Res(z − i )∂S α ii∫1(2.5)=f(z)Ψ(z) dz − ∑ Res(z i )Ψ(z i )2πi ∂S α iand(2.6) h∞∑n=−∞f(nh) = 1 ∫( πz)f(z)π cot dz − ∑ 2πi ∂S αhiRes(z i )π cotHere z + i and z + i represent the poles above and below the real line.( πzi).h2.2. Our working notation. We will work with the following quantities for h > 0.(1) The integral: If := ∫ 1−1 f.(2) The tanh-sinh approximation:N∑ π cosh(nh)( ( π))T h,N f := h2 cosh 2 ( π2 sinh(nh)) f tanh2 sinh(nh) .n=−N(3) The N-th approximation error:(4) The approximation limit:E h,N f := (I − T h,N ) f.T h f = lim T h,N f,N−→∞which will be shown to exist in lemma 3.1.(5) The limit error:E h := (I − T h ) f.Then I, T h,N , E h,N and are bounded linear functionals on H 2 . So are T h , andE h once we show they exist.3. The associated space G 2Along with H 2 , it is helpful to use the same change of variable as in (1.3) todefine a corresponding space of functions, [4]. Precisely, we let G 2 be the set offunctions of the formψ ′ (w)f (ψ(w)) f ∈ H 2 and ψ as in (1.3).Assumptions. We assume ψ maps region A onto the unit disk, that functions inG 2 are analytic on A, and are defined almost everywhere on ∂A.Letting ̂f(w) := ψ ′ (w)f (ψ (w)), we can induce an inner product in G 2 by〈 ̂f, ĝ〉 G 2 := 〈f, g〉 H 2 = 1 ∫f(z)g(z) |dz|2π |z|=1= 1 ∫(3.1)f(ψ(w))g(ψ(w))|ψ ′ (w) dw|2π ∂A= 1 ∫̂f(w)ĝ(w) |dw/ψ ′ (w)|.2π∂A