I have a messy expression that contains the incomplete gamma function
and want to examine its behavior in certain limits. maxima doesn't
seem to have these functions defined, but they are easy to define
explicitly as integrals. Unfortunately, things don't work the way I
want. Here's a trivial test code:
assume(x>0);
f(d,x):=integrate(z**(d-1)*exp(-z),z,0,x);
f(1,x);
taylor(f(d,x),d,1,1);
and here's what I get when I run it:
(%i30) assume(x>0);
(%o30) [redundant]
(%i31) f(d,x):=integrate(z**(d-1)*exp(-z),z,0,x);
d - 1
(%o31) f(d, x) := integrate(z exp(- z), z, 0, x)
(%i32) f(1,x);
- x
(%o32) 1 - %e
(%i33) taylor(f(d,x),d,1,1);
! !
x ! x !
/ ! / !
[ d - 1 - z ! [ d - 1 - z !
(%o33)/T/ I z %e dz! + (I z %e log(z) dz! )(d -
1)
] ! ] !
/ ! / !
0 ! 0 !
!d = 1 !d = 1
+ . . .
This is correct, but I'd rather have the setting d=1 brought into the
integrals, and in the case of the 1st term, have the integral done.
(The second integral can be done exactly, although the result contains
yet more special functions, and I don't think that maxima will be able
to do this).
Any suggestions?
David