Integral of sin(x)/x gives false result.

[sagetrac-benreynwar]

> eq = sin(x)/x
> integrate(eq, x, -1e-6, 1e-6)
-3.14159065359

I expected an answer of about 2e-6.

Upstream: Reported upstream. Developers acknowledge bug.

Component: calculus

Keywords: maxima, integration

Stopgaps: todo

Issue created by migration from https://trac.sagemath.org/ticket/11164

[kcrisman]

Upstream: Fixed upstream, in a later stable release.

[kcrisman]

comment:1

In the future, be sure to pick a component (for instance, calculus or symbolics); that will help people find it more easily.

There are two problems here.

• The actual problem - the report unfortunately has a typo.

sage: integrate(eq,x,-1e-6,1e-6)
-3.14159065359

This is definitely a problem. However, it's fixed in the latest Maxima. Here is the old one, in Sage:

Maxima 5.22.1 http://maxima.sourceforge.net
using Lisp ECL 10.4.1
<snip>
(%i2) keepfloat:true;
(%o2)                                true
(%i3) integrate(sin(x)/x,x,-1.00000000000000e-6,1.00000000000000e-6)
;
(%o3)                         - 3.141590653589793

And here is 5.24.0. I assume this isn't a problem of which Lisp we are using?

Maxima 5.24.0 http://maxima.sourceforge.net
using Lisp SBCL 1.0.24
(%i3) keepfloat:true;
(%o3)                                true
(%i8) integrate(sin(x)/x,x,-1.00000000000000e-6,1.00000000000000e-6);

rat: replaced 9.999999999999999e-7 by 1/1000000 = 9.999999999999999e-7

rat: replaced -9.999999999999999e-7 by -1/1000000 = -9.999999999999999e-7

rat: replaced -9.999999999999999e-7 by -1/1000000 = -9.999999999999999e-7

rat: replaced 9.999999999999999e-7 by 1/1000000 = 9.999999999999999e-7
                                 %i                                %i
(%o8) %i gamma_incomplete(0, - -------) - %i gamma_incomplete(0, -------)
                               1000000                           1000000

• A problem revealed, but also fixed in the latest. Old:

(%i4) integrate(sin(x)/x,x,-1e-6,1e6);

Maxima encountered a Lisp error:

 #<a FLOATING-POINT-OVERFLOW>

Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.

New:

(%i10) integrate(sin(x)/x,x,-1e-6,1e6);

rat: replaced 9.999999999999999e-7 by 1/1000000 = 9.999999999999999e-7

rat: replaced -9.999999999999999e-7 by -1/1000000 = -9.999999999999999e-7

rat: replaced -1000000.0 by -1000000/1 = -1000000.0

rat: replaced 1000000.0 by 1000000/1 = 1000000.0
                                                                       %i
                                              %i gamma_incomplete(0, -------)
         %i gamma_incomplete(0, 1000000 %i)                          1000000
(%o10) - ---------------------------------- - -------------------------------
                         2                                   2
                                  %i
       %i gamma_incomplete(0, - -------)
                                1000000    %i gamma_incomplete(0, - 1000000 %i)
     + --------------------------------- + ------------------------------------
                       2                                    2

We'd need doctesting to test both of these once we upgrade Maxima.

[sagetrac-dsm]

add doctest to confirm fix

[sagetrac-dsm]

comment:2

Attachment: trac_11164_verify_integration_sinx_x.patch.gz

Might as well doctest and close.

[kcrisman]

comment:3

I may have misstated whether this is "fixed".

sage: integrate(sin(x)/x,x,-1e-6,1e-6)
I*Ei(-1/1000000*I) - I*Ei(1/1000000*I)
sage: integrate(sin(x)/x,x,-1e-6,1e-6).n()
3.14159465358979
sage: numerical_integral(sin(x)/x,-1e-6,1e-6)
(1.9999999999998889e-06, nan)

The graph makes it very obvious the second (numerical) answer is correct. Unsurprisingly, the other answer is off by exactly pi...

Presumably the first answer is "correct" in the sense that it is an alternate representation of the Si sine integral function. But I think there are some branch issues - Ei(I*x) only has an alternate formula for x>0.

So we still need to do something with this. Or at the very least warn about branches. But really, sin(x)/x shouldn't have to worry about this!

[sagetrac-dsm]

comment:4

You're right, of course. It's silly to start with sin(x)/x and then generalize to something which doesn't reduce to the expected value, but far, far sillier to add a doctest enshrining the goofy behaviour as the expected. :^)

Does this work on the current maxima trunk now?