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tricky_integrals - Last revised 2008-Apr-01 Integrals with...

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Last revised: 2008-Apr-01 Integrals with Singularities or Discontinuous Derivatives Numerical integration algorithms such as Simpson’s 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 0 - 1 ( - x ) f ( x ) dx + Z 1 0 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 x 1 /n with n an integer greater than one, a simple variable change can usually be found to convert the integrand to a smooth one. For
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