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Double Exponential

Double Exponential

Quadrature Formula

(-1, 1)

\[\begin{eqnarray} \frac{\pi}{2} h \sum_{i=-n}^{n} \frac{ f(\tanh(\pi \sinh(h * i) / 2)) \cosh(i h) }{ \cosh^{2}(\pi h / 2 \sinh(h * i)) } \end{eqnarray}\]

(0, \infty)

\[\begin{eqnarray} \frac{\pi}{2} h \sum_{i=-n}^{n} f(\exp(\pi \sinh(h * i) / 2)) \cosh(i h) \exp(\pi \sinh(h * i) / 2) \end{eqnarray}\]

(-\infty, \infty)

\[\begin{eqnarray} \frac{\pi}{2} h \sum_{i=-n}^{n} f(\sinh(\pi \sinh(h * i) / 2)) \cosh(i h) \cosh(\pi \sinh(h * i) / 2) \end{eqnarray}\]

Converting finite interval to (-1, 1)

\[\begin{eqnarray} \int_{a}^{b} f(x) \ dx & = & \int_{-1}^{1} f \left( \frac{ b - a }{ 2 } \left( t + \frac{ b + a }{ b - a } \right) \right) \ dt \end{eqnarray}\]

Reference