> f(x):= asin(sqrt(1-x^2)/(1+eps));
> taylor(f(x),x,0,2);
(infinite loop)
The problem appears to be in the default gcd; you can get around this by
changing gcd routine:
gcd:'spmod$
> g(x):=asin(sqrt(1-x^2));
> taylor(g(x),x,0,2);
> "Taylor encountered an unfamiliar singularity in ABS(X)"
This is a bug in Taylor. To get around it, first expand the inner
expression, then plug it into asin:
inner: taylor(sqrt(1-x^2),x,0,4);
taylor(asin(inner),x,0,4);
In general, you have to be careful about the orders of expansion -- you
may need more terms in the inner expression than the whole expression.
That is the sort of thing Taylor is supposed to do for you, but
obviously there is a problem in this case.
-s
PS I am ignoring the subtler issues around the sign of sqrt etc.