Hi everyone,
I've just discovered Maxima a few days ago, and therefore I'm far from being
an experienced user... I have to say Maxima really is fantastic, I just love
it !
However I've come across the following problem, involving residues :
(%i1) set_display('ascii) $
domain : complex $
declare(N, integer) $
H : 1 / (x*(x - x0)*(1 - x^N/y)) $
(%i5) residue(H, x, 0);
y
(%o5) - ------------
N
x0 y - x x0
Why oh why does the residue of H(x) at 0 depend on x ? It actually turns out
that the result is correct, when x is replaced with 0 :
(%i6) subst(x=0, %);
1
(%o6) - --
x0
I kind of guess that the problem is related to Maxima being unable to
perform the full partial fraction decomposition, although 0 is a pole of
order 1 (not even a multiple pole) :
(%i7) partfrac(H, x);
y y
(%o7) ----------------------- - ----------------
N N
(x - x0) (x0 y - x x0) x (x0 y - x x0)
There might be something I don't know about Maxima, and the way it solves
problems... It's just a bit disturbing that the residue of H(x) depends on
x... I think this is not something the user might reasonably expect.
At last, a related question / suggestion :
(%i8) declare(k, integer) $
Hk : x^(k-1) / ((x - x0)*(1 - x^N/y)) $
(%i10) residue(Hk, x, 0);
(%o10) 0
Here I would have thought Maxima would ask if k is positive or not...
As a matter of fact, the answer is 0 for all positive k, but not for
negative k (including k=0, where the result was found earlier...). Does
Maxima provide "conditional results", with various expressions depending on
the value of a parameter ? I appreciate that Maxima can't do everything, and
that users have to know what they are doing in the first place... But
sometimes it helps to be warned that things are a bit more subtle than they
seem (e.g. someone who looked a bit quickly would have thought that all
residues are 0, including for k<=0).
Funnily enough I've just noticed the following :
(%i17) declare(p, integer) $
residue(subst(k=+p, Hk), x, 0);
residue(subst(k=-p, Hk), x, 0);
(%o18) 0
y
(%o19) - -------------------
p N + p
x x0 y - x x0
Again, maybe I'm really missing something about how Maxima works, but I
would not expect the result to be different depending on the sign affected
to a parameter (which can be positive or negative itself)...
Many thanks for any comments, and to anyone who can shed some light on
Maxima's philosophy !
David.