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Re: Problem with change of variables in an integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg131582] Re: Problem with change of variables in an integral
*From*: Youngjoo Chung <ychung12 at gmail.com>
*Date*: Sun, 8 Sep 2013 03:06:54 -0400 (EDT)
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Dear Alexei,
Integrate[...] in Map[...] will be evaluated first before Map can apply ReplaceAll[...], so nothing is done by Map.
There is a typo in
Integrate[E^(i \[Phi])/(1 + (E^(i \[Phi]) r)^3), {r, 0, \[Infinity]}]
i needs to be changed to I. Then it will produce the same result.
Transforming integrals by change of variables is not straightforward in Mathematica, especially if the integral can be evaluated in closed form. One method involves wrapping the integral with Hold or HoldForm so that evaluation is not done. In addition, simple replacement of the integration variables will not work and the integrand must be divided by the Jacobian corresponding to the variable transformation.
Another method is to write a customized code based on the box language of Mathematica to interpret the 2-D form of integrals, which does not evaluate the integrals but does so only when asked. This is what is done in the SymbolicComputing package (http://symbcomp.gist.ac.kr, WTC 2011). The commands are:
<<SymbolicComputing`
SCTransInt[\[Integral]f[z] \[DifferentialD]z,
TransVar -> {z, r, z == r E^(I \[Phi])},
ReplVar -> {r, 0, \[Infinity]}] /. \[Phi] -> 2 \[Pi]/3
It is to be noted that once the package is loaded, integrals, products and sums in 2-D form are not evaluated. Evaluation can be done using the functions SCEvalInt, SCEvalProd, and SCEvalSum, respectively. Kernel functions Integrate, Product, and Sum are the same and can be used for immediate evaluation.
Sincerely,
Youngjoo Chung
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