[Maxima] Further work on $specint
drdieterkaiser at web.de
Thu Jun 19 17:37:48 CDT 2008
I continue the work on $SPECINT. On the one hand I try to improve the existing
algorithm, on the other hand I try to extend $SPECINT further.
First I added an algorithm to calculate more general integrals of the typ
t^v*(a+t)^w. The general result of this integral can be expressed in terms of
the Hypergeometric U function. In special cases this can be transformed to a
Gammaincomplete function or a sum of Hypergeometric 1F1 functions. For w=0 we
get the result in terms of a Gamma function. This case is allready implemented
For v=0 and w<>0 we get results in terms of the Gammaincomplete function:
Two more general cases which transform to 1F1 functions which simplifies to Erf
Next I have implemented a first version to integrate functions containing the
Unit Step function. I used the existing function unit_step from orthopoly. The
algorithm works in principle, but I have to do further work to do it more
generally including sums of Unit Step functions.
Here some examples with the new code:
The following result can be shown to simplify to an Erf function:
For integrals with the Unit Step function it would be also nice to have an
algorithm to integrate general sums like (sum unit_step(t-n*a),n,0,inf).
A next step to complete the integration of the tabulated integrals would be to
integrate expressions containing the Abs function. The algorithm should be not
to difficult to implement. I am working on it.
It would be nice when the Hypergeometric functions and other special functions
would be simplifying functions. I think it is not a good idea to extend $SPECINT
to give more simple and equivalent results. A more natural way would be to have
simplifying routines for the functions themselfs.
An example is the following integral (with the new algorithm for t^v*(a+t)^w):
This result can be expressed in terms of the Erfc function with the known
special value for the Gammaincomplete function:
This could be the result of a simplifying routine for the Gammaincomplete
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