Hello Ted,
I had a further look into the code of the integrator.
I have found a method to generalize the integrator even more. I have implemented
a routine to extend the method of integration by parts to special functions.
Here some first result:
The integration of the function itself:
(%i6) integrate(expintegral_ei(x),x);
(%o6) x*expintegral_ei(x)-%e^x
A result obtained by the method "derivative-divides" extended to special
functions:
(%i7) integrate(expintegral_ei(x^2)*x,x);
(%o7) (x^2*expintegral_ei(x^2)-%e^x^2)/2
Here two examples obtained with the method "integration by parts":
(%i8) integrate(expintegral_ei(x)*x,x);
(%o8) x^2*expintegral_ei(x)/2-(x-1)*%e^x/2
(%i9) integrate(expintegral_ei(x)*(a*x^2+b*x+c),x);
(%o9) (a*x^3/3+b*x^2/2+c*x)*expintegral_ei(x)
-(2*a*x^2+(3*b-4*a)*x+6*c-3*b+4*a)*%e^x/6
Now I am working on the code to extend the integrator to expressions like
exp(x)/x. With this last extension Maxima could solve a lot of integrals
involving special functions.
Dieter Kaiser
-----Urspr?ngliche Nachricht-----
Von: woollett at charter.net [mailto:woollett at charter.net]
Gesendet: Donnerstag, 30. Oktober 2008 18:52
An: drdieterkaiser at web.de
Cc: maxima mailing list
Betreff: Re: [Maxima] exponential integral as solution of ode?
On Oct. 29, Dieter Kaiser wrote:
>A fast implementation I have done today would give the integrals of the
>special
>functions themselves. Here are some examples I have got today with an
>extension
>to the routines of sinint:
>
>(%i14) integrate(expintegral_ei(x),x);
>(%o14) x*expintegral_ei(x)-%e^x
>--------------
>Perhaps such an extension would be a first and interessting step to give
>more
>support to the special functions.
Adding support for returning the indefinite integrals
of special functions would greatly increase
Maxima's ability to solve differential equations
frequently met in applications.
Also, thanks for your work to get reliable
values for complex arguments.
Ted Woollett