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Why does this non-zero integral evaluate to 0?

asked 2023-03-29 09:43:21 +0100

cs89 gravatar image

I want to perform formal computations using SageMath, which involve integrals of trigonometric functions. I noticed some inconsistencies and narrowed my issues down to the following basic problem.

Why does the following source code return 0?

var('n t')
assume(n, 'integer')
integrate(cos(2 * pi * n * t), t, 0, 1)

While I agree that the integral is zero when n is non-zero, mathematically, the integral is worth 1 when n is zero. Up to my understanding, I only tell SageMath that n is an integer here, so it should handle the case n = 0 also.

I checked that integrate(cos(2 * pi * 0 * t), t, 0, 1) correctly returns 1, so SageMath seems to be aware of this possibility, but somehow misses the fact that n could be 0.

What am I missing?

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kcrisman gravatar imagekcrisman ( 2023-03-31 22:54:29 +0100 )edit

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answered 2023-03-31 23:09:52 +0100

kcrisman gravatar image

You can reproduce this with the following Maxima code:

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

which will return 0. As one of the Maxima devs points out in his comment at the SO question about the same topic:

Although there are places in the code where Maxima will ask if a variable is zero or nonzero, these are triggered only if a corresponding special case is detected; Maxima doesn't try to analyze results to see if there are cases which aren't covered, such as n = 0 in your example.

In this case, a little further investigation reveals that the antiderivative found is wrong for $n=0$:

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

which obviously doesn't work in that case. In fact, we get this result even if n is not an integer!

I've opened Maxima bug report 4125 for this. Thanks for reporting!

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Also note Sympy does this correctly, though Sage can't interpret its definite integral (it does do the antiderivative correctly). See https://github.com/sagemath/sage/issu... for more progress and tracking.

kcrisman gravatar imagekcrisman ( 2023-03-31 23:19:22 +0100 )edit

Update: Maxima devs don't consider a removable discontinuity a bug, but the integer bit remains.

kcrisman gravatar imagekcrisman ( 2023-03-31 23:58:04 +0100 )edit

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Asked: 2023-03-29 09:43:21 +0100

Seen: 2,399 times

Last updated: Mar 31 '23