[Maxima] Further work on $specint

Dieter Kaiser drdieterkaiser at web.de
Sun Jun 1 13:40:25 CDT 2008

After some further work on $SPECINT I have reported that one problem of the 99
examples of EqWorld is open and some further examples have to be verified more

Here is an example with sinh(2*sqrt(k*t)) which is correct (A&S 29.3.79) after
correcting the code of $SPECINT (The original code gives the result 0):

(%i14) radcan(specint(%e^(-s*t)*sinh(2*sqrt(t*k))/sqrt(%pi*k),t));
(%o14) %e^(k/s)/s^(3/2)

Now we try sqrt(t)*sinh(2*sqrt(k*t)). 

(%i17) radcan(specint(%e^(-s*t)*sqrt(t)*sinh(2*sqrt(t*k)),t));
(%o17) (sqrt(%pi)*erf(sqrt(k)/sqrt(s))*s*%e^(k/s)

The answer of Maxima looks very good. But the tabulated result of Eqworld  has a
different sign in the last term. The result of EqWorld would be 
...-2*sqrt(k)*sqrt(s). I think EqWorld has a wrong sign.

Now we try sinh(2*sqrt(k*t))^2/sqrt(t) and cosh(2*sqrt(k*t))^2/sqrt(t):

(%i24) radcan(specint(%e^(-s*t)*sinh(2*sqrt(t*k))^2/sqrt(t),t));
(%o24) sqrt(%pi)*(%e^(4*k/s)-1)/(2*sqrt(s))

(%i25) radcan(specint(%e^(-s*t)*cosh(2*sqrt(t*k))^2/sqrt(t),t));
(%o25) sqrt(%pi)*(%e^(4*k/s)+1)/(2*sqrt(s))

The two examples have a factor %e^(4*k/s) in the result. 
For these cases EqWorld gives a factor %e^(k/s) too. I think EqWorld is wrong.

There remains the following open problem:

(%i4) radcan(specint(%e^(-s*t)*erf(sqrt(a/t)/2),t));
(%o4) (sqrt(a)*sqrt(s)*sqrt(4-a*s)+4*asin(sqrt(a)*sqrt(s)/2))/(4*s)

Maxima gives a result, but the tabulated result is 1/s*(1-%e^(-sqrt(a*s))). I
think the result of Maxima is wrong and we have an error in the algorithm with
an argument of the form sqrt(a/t) for the Erf function.

There are a lot of integrals Maxima only give a correct noun form (after some
extensions of the code). So it would be a nice work to extend the algorithm of
$SPECINT systematically. Some extensions are allready implemented with the
changes given up to now and presented as attachements to a bug report on

To get even more examples I have collected additionally about 130 integrals
which are tabulated by A&S.

Dieter Kaiser

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