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Unformatted text preview: Last revised: 2008-Apr-01 Integrals with Singularities or Discontinuous Derivatives Numerical integration algorithms such as Simpsons rule are designed to work with smooth integrands, because they assume that the function is locally a polynomial. If the integrand has discontinuous derivatives or poles or a branch point in the integration region, such integration rules will generally do poorly. However, one can convert the original integral into a mathematically equivalent one that is smooth in this sense. Here are some strategies (check Numerical Recipes and the Hjorth-Jensen notes for others): If there is a discontinuous derivative somewhere in the integrand, then split the integral into two integrals at the discontinuity. For example, if there is an absolute value, then Z 1- 1 | x | f ( x ) dx = Z- 1 (- x ) f ( x ) dx + Z 1 x f ( x ) dx , (1) if f ( x ) is smooth. (One can combine the integrals after taking x - x in the first.) If there is a branch point, such as...
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