Maxima 5.29.1 in ECL. Apparently the limit is actually 1, see https://groups.google.com/forum/#!topic/sage-support/y9UnyWbUagY
(%i2) y:gamma(x+1/2)/(sqrt(x)*gamma(x));
1
gamma(x + -)
2
(%o2) ----------------
sqrt(x) gamma(x)
(%i3) limit(y,x,0);
(%o3) 0
Peter Bruin
2014-05-18
It seems to me that the limit as x -> 0 is correctly computed as 0.
The question originally asked in the discussion linked to, however,
was about the limit as x -> infinity.
For any real number a, Stirling's approximation shows that the
function
f: gamma(x + a)/(x^a*gamma(x));
tends to 1 as x -> infinity. This is also what
limit(x, a, inf);
gives you (after asking a few questions). However, for specific
rational but non-integral values of a the result returned by Maxima
seems to be either 0 or inf. Besides the above example (using
limit(y, x, inf)) one also gets, for example,
(%i23) f: gamma(x - 2/5)/(x^(-2/5)*gamma(x)); 2/5 2 x gamma(x - -) 5 (%o23) ----------------- gamma(x) (%i24) limit(f,x,inf); (%o24) 0
I tried to do a bit of debugging. It seems that the limit (in the
case a = 1/2, say) is computed via limit(g, x, inf), where
g: 1/sqrt(x)*exp(x*log(2*x-1)-(x-1/2)*log(x-1)-1/2)/2^x;
as obtained by Stirling's approximation. Indeed, trying to compute
this limit also yields infinity instead of the correct value 1.
When we instead simplify the expression for g to
g1: 1/sqrt(x)*exp(x*log(x-1/2)-(x-1/2)*log(x-1)-1/2);
the limit is still computed as infinity, but this time it takes
several minutes. I don't know if the slowness is related to the
incorrect answer. Finally, further simplifying g1 to
g2: exp(x*log(x-1/2)-(x-1/2)*log(x-1)-1/2-log(x)/2);
and computing limit(g2, x, inf) does instantaneously return the
correct answer 1.
(All of the above is in Maxima 5.33.0 on GCL.)
Thanks,
Peter Bruin
Peter Bruin
2014-05-20
I finally discovered the reason for this bug: the series expansion
step in Gruntz's algorithm (see his thesis [1]) is not implemented
correctly. On page 49, it is stated that when expanding a logarithm,
log w must be replaced by h. In Gruntz's Maple implementation
(appendix of [1]), he does this by means of a magical declaration
ln(W) := e3;
that is somehow picked up by Series()
.
In the Maxima implementation, the substitution log w -> h is only done
after expanding the whole expression. The problem is that by time,
the log w may have been transformed, e.g. because it appears inside an
exponential.
The patch limit-rewrite-logs.patch
[see message below] fixes this,
at least for "pure exp-log functions" (the ones for which Gruntz's
algorithm was designed) by doing a suitable substitution on relevant
subexpressions of the form log(f(x)) before the series expansion.
The patch currently keeps the existing substition after the series
expansion because (1) if the expression involves other transcendental
functions whose Taylor expansion involves logarithms, we still want to
transform those if we see them, and (2) not doing this substitution
led to errors even in cases where it is apparently unnecessary,
e.g. in the test
integrate(x^6/(1 + x + x^2)^(15/2), x, 0, inf);
I hope this is an acceptable fix for this bug.
[1] D. Gruntz, On computing limits in a symbolic manipulation
system. Ph.D. thesis, ETH Zürich, 1996,
http://e-collection.library.ethz.ch/view/eth:40284
Peter Bruin
2014-05-20
Here is a slightly simplified version of the patch. [Edit: simplified once more]
Robert Dodier
2014-05-27
Robert Dodier
2014-05-27
Fixed by commit [b0579c08a] (applied patch).