#4125 incorrect integral for cosine when coefficient is zero, potentially

[kcrisman]

integrate(cos(2 * %pi * n * t), t, 0, 1);

yields

sin(2*%pi*n)/(2*%pi*n)

which clearly doesn't work if n is zero.

First reported at https://ask.sagemath.org/question/67139/why-does-this-non-zero-integral-evaluate-to-0/ and https://stackoverflow.com/questions/75869399/why-does-this-non-zero-integral-evaluate-to-0-in-sagemath

[kcrisman]

For reference, the antiderivative is what is wrong:

integrate(cos(2 * %pi * n * t),t);
sin(2*%pi*n*t)/(2*%pi*n)

[kcrisman]

See also downstream at https://github.com/sagemath/sage/issues/35406

[Stavros Macrakis]

This isn't a bug. There is a removable singularity at n=0, and the limit gives the correct result:

(%i1) integrate(cos(2*%pi*n*t),t,0,1);
                                 sin(2 %pi n)
(%o1)                            ------------
                                   2 %pi n
(%i2) limit(%,n,0);
(%o2)                                  1
(%i3) integrate(cos(2*%pi*0*t),t,0,1);
(%o3)                                  1

[kcrisman]

Probably debatable (the accumulation function when n=0 clearly doesn't have any singularities (discontinuities?), removable or otherwise), but here is the actual original report:

declare(n, integer);
integrate(cos(2 * %pi * n * t), t, 0, 1);

This returns zero, and that certainly is a bug.

[Stavros Macrakis]

Agreed. Maxima should ask whether n is zero if n is declare integer. I've reopened the bug. Please be sure to include the full context in bug reports -- it's pretty important!

[kcrisman]

Great, thanks.

[kcrisman]

Context - agreed! I just noted that Sympy considers the cases n=0 and not zero separately, so I considered the Maxima behavior a bug, and was trying to declutter.

[Devin Greene]

I'd add that this issue may be having some downstream effects on
packages.

For example:

(%i1) load(fourie)$ 

(%i2) declare(n,integer)$

(%i3) fourier(cos(x),x,%pi);
(%t3)                               a  = 0
                                     0

(%t4)                               a  = 0
                                     n

(%t5)                               b  = 0
                                     n

(%o5)                           [%t3, %t4, %t5]
(%i6) 

cos(x) is certainly not zero over the interval [-%pi,%pi] !