Tanh-sinh quadrature

Remarks

Clause 1 is the change of variables formula, since $\varphi' > 0$ and $\varphi(t) \to \pm 1$ as $t \to \pm\infty$. Clause 2 follows from $\cosh u \ge \frac12 e^{\abs{u}}$, which gives $\cosh^{-2}\big(\frac{\pi}{2}\sinh t\big) \le 4 e^{-\pi \sinh\abs{t}}$. So the transformed integrand decays double exponentially, and the cut-off at $\abs{t} \le Nh$ loses almost nothing; this is why the method is also called double exponential quadrature. The trapezoidal rule on $\R$ is extremely accurate for functions analytic in a strip around the real axis. For integrands analytic in a suitable region around $(-1, 1)$, the error of $T_{h,N}$ with a good choice of $h$ decreases roughly like $e^{-cN/\log N}$ for some $c > 0$. Example: for $f(x) = (1 - x^2)^{-1/2}$, which is unbounded, the transformed integrand is $\frac{\pi}{2} \cosh t / \cosh\big(\frac{\pi}{2} \sinh t\big)$, still double exponentially decaying, and its integral is $\pi$. The method was introduced by Hidetosi Takahashi and Masatake Mori in 1974.

Going deeper

For a student

The method is robust against endpoint singularities, which defeat Gaussian rules, and it reuses all old samples when the step $h$ is halved. For these reasons it is a standard choice in arbitrary-precision computation, where hundreds of correct digits are needed. It needs care near the endpoints in floating-point arithmetic, since $\varphi(kh)$ is extremely close to $\pm 1$ and $1 \mp \varphi(kh)$ should be computed directly. Related double exponential substitutions treat integrals over $[0, \infty)$ and $\R$.