On Oct. 24, 2011, Dieter Kaiser wrote in a bug fix message:
-----------------------------------------
>Bugs item #3427581, was opened at 2011-10-23 19:39
>Message generated for change (Comment added) made by crategus
>[snip]
>Message:
>This bug has been fixed in bessel.lisp revision 23.10.2011.
>
>(%i1) imagpart(bessel_i(1,%i*x));
>(%o1) -%i*bessel_i(1,%i*x)
>
> Nevertheless, the reported example is not solved:
>
>(%i2) quad_qag(imagpart(bessel_i(1,%i*x)),x,1,3,3,limit=700);
>(%o2) quad_qag(-%i*bessel_i(1,%i*x),x,1,3,3,epsrel = 1.e-8,epsabs = 0.0,
> limit = 700)
>
>The reason is a small imaginary part because of rounding errors:
>
>(%i3) imagpart(bessel_i(1,%i));
>(%o3) -%i*bessel_i(1,%i)
>(%i4) rectform(%),numer;
>(%o4) .4400505857449336-2.694443716532522e-17*%i
>
>For other Bessel functions Maxima knows pure real or imaginary results and
>cuts off the rounding errors accordingly. This can be added for the
>bessel_i function too.
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nint.mac is now using the new bessel.lisp.
Until the rounding error problem of bessel_i is fixed, we can use
a workaround suggested by you (Dieter Kaiser), namely to use
a float number instead of an integer for the order of the bessel
function.
-------------------
(%i1) load(nint);
(%o1) "c:/work2/nint.mac"
(%i2) quad_qag(imagpart(bessel_i(1,%i*x)),x,1,3,3,limit=700);
(%o2) quad_qag(-%i*bessel_i(1,%i*x),x,1,3,3,epsrel = 1.0E-8,epsabs = 0.0,
limit = 700)
(%i3) quad_qag(imagpart(bessel_i(1.0,%i*x)),x,1,3,3,limit=700);
(%o3) [1.0252496414599,1.1382557579374679E-14,31,0]
(%i4) quad_qag(imagpart(bessel_i(1.0,%i*x)),x,1,10^4,3,limit=700);
(%o4) [0.77229384691138,1.1418381427522419E-12,31713,0]
(%i5) quad_qag(imagpart(bessel_i(2.0,%i*x)),x,1,3,3,limit=700);
(%o5) [8.2026594591804487E-17,9.1067813947414708E-31,31,0]
(%i6) quad_qag(imagpart(bessel_i(2.0,%i*x)),x,1,10^4,3,limit=700);
(%o6) [-2.2013154155922133E-16,5.7377810005552024E-15,43369,1]
(%i7) quad_qag(imagpart(bessel_i(3.0,%i*x)),x,1,3,3,limit=700);
(%o7) [-0.28287409015192,3.1405332795655258E-15,31,0]
(%i8) quad_qag(imagpart(bessel_i(3.0,%i*x)),x,1,10^4,3,limit=700);
(%o8) [-0.98790703708805,1.0431047896924372E-12,31713,0]
------------------------------
Ted