behavior of 'taylor' when Taylor series is undefined
Subject: behavior of 'taylor' when Taylor series is undefined
From: Barton Willis
Date: Tue, 18 Dec 2012 18:13:59 +0000
> I wonder what should be the behavior of 'taylor' when no Taylor series exists.
Taylor sometimes gives an error when it simply doesn't know what to do:
(%i6) taylor(bessel_j(%pi,x),x,0,5);
taylor: encountered an unfamiliar singularity in: bessel_j(%pi,x)
The problem isn't that bessel_j(%pi,x) doesn't have a power series representation about zero--it's just that Maxima doesn't handle this case.
Converting bessel_j(%pi,x) to a hypergeometric form (using an experimental function), allows Maxima to find a series representation:
(%i11) to_hypergeometric(bessel_j(%pi,x));
(%o11) (hypergeometric([],[%pi+1],-x^2/4)*x^%pi)/(2^%pi*gamma(%pi+1))
(%i12) taylor(%,x,0,5);
(%o12)/T/ x^%pi/(2^%pi*gamma(%pi+1))-(x^%pi*x^2)/((4*%pi+4)*2^%pi*gamma(%pi+1))+(x^%pi*x^4)/((32*%pi^2+96*%pi+64)*2^%pi*gamma(%pi+1))+...
And taylor sometimes gives an error when a series representation (possibly noninteger powers) doesn't exist:
(%i7) taylor(abs(x),x,0,3);
taylor: encountered an unfamiliar singularity in: abs(x)
The function to_hypergeometric has been rattling around my machine for a long time...
--Barton