Because I have implemented some more code for the Incomplete Beta functions I
had again a look at the beta function.
I would like to suggest the following additional functionality:
1. Expand beta(a+n,b) and beta(a-n,b) for n an integer
The expansion depends on the flag $beta_expand. The expressions beta(a,b+n)
and beta(a,b-n) are simplified accordingly.
2. Set the operator property symmetric: beta(a,b) = beta(b,a)
With this additional functionality expressions with beta functions would
simplify much better.
Two examples:
(%i5) beta(a,b)-beta(b,a);
(%o5) 0
(%i6) beta(a+1,b)/beta(a,b+1),beta_expand:true;
(%o6) a/b
There is only one change in the testsuite. Example 72 in rtestint.lisp fails.
The result of the integral in this example is beta(5/7,2/7) which simplifies to
%pi/sin(5*%pi/7).
After the extension to give beta the property symmetric the expression
beta(5/7,2/7) simplifies first to beta(2/7,5/7). Now we get the result
%pi/sin(2*%pi/7). But both results are equivalent and evaluate to the same
numerically value.
The share_testsuite would have no problems with the additional functionality.
So, what do you think? Should we give the beta function the operator property
symmetric?
Dieter Kaiser