Incomplete Gamma fails for negative integer arguments
Subject: Incomplete Gamma fails for negative integer arguments
From: Dieter Kaiser
Date: Sun, 26 Oct 2008 19:53:34 +0100
The numerical algorithm of gamma_incomplete(n,z) do not work for an argument n=0
or n a negative integer where realpart(z)<0.
The reason is that for these cases the series expansion for the numerically
evaluation does not work. Maxima gets a term in the expansion with a division by
zero.
I do not know why I have omitted to test these values.
Here are the results:
(%i18) gamma_incomplete(0,0.5);
Maxima encountered a Lisp error:
Error in PROGN [or a callee]: Zero divisor.
Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
n negative integer, but z positive. This works, because Maxima uses an expansion
in continued fractions.
(%i19) gamma_incomplete(-1,0.5);
(%o19) 0.65328772464911
n negative integer and z<0. Error. The series expansion fails. (Remark: An
expansion in continued fractions would fail too. It does not converge.)
(%i20) gamma_incomplete(-1,-0.5);
Maxima encountered a Lisp error:
Error in PROGN [or a callee]: Zero divisor.
Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i21) gamma_incomplete(-2,-0.5);
Maxima encountered a Lisp error:
Error in PROGN [or a callee]: Zero divisor.
Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
Two possible ways to solve the problem:
1. Improve the series expansion for the special cases.
Might not be possible.
2. Using expintegral_e(n,z) for these special values.
I am working on the problem.
Dieter Kaiser