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Double exponential formulas for numerical indefinite integration. (English) Zbl 1038.65018
Summary: We derive a formula for indefinite integration of analytic functions over $$(-1,s)$$ where $$-1 < s < 1$$, by means of the double exponential transformation and the Sinc method. The integrand must be analytic on $$-1 < x < 1$$ but may have a singularity at the end points $$x = \pm 1$$. The error of the formula behaves approximately as exp$$(-c_1N/{\log} c_2N)$$ where $$N$$ is the number of function evaluations of the integrand. This error term shows a much faster convergence to zero when $$N$$ becomes large than that of the known formula by S. Haber [Math. Comput. 60, No. 201, 279–296 (1993; Zbl 0795.65008)]. Also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over $$(0,s), 0<s<{\infty}$$ and over $$(-{\infty},s)$$, $$-\infty < s < +\infty$$. Several numerical examples indicate high efficiency of the formulas.

##### MSC:
 65D30 Numerical integration 41A55 Approximate quadratures
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##### References:
  Haber, S., Two formulas for numerical indefinite integration, Math. comp., 60, 279-296, (1993) · Zbl 0795.65008  M. Mori, M. Muhammad, A method for numerical evaluation of indefinite integrals by the double exponential transformation, Proceedings of 2002 Symposium on Applied Mathematics, Ryukoku University, Ohtsu, December 19-21, 2002 (in Japanese).  Sugihara, M., Optimality of the double exponential formula – functional analysis approach, Numer. math., 75, 379-395, (1997) · Zbl 0868.41019  Takahasi, H.; Mori, M., Double exponential formulas for numerical integration, Publ. res. inst. math. sci., 9, 721-741, (1974) · Zbl 0293.65011
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