Indefinite integration for piecewise functions

[sagetrac-pbutler]

It would be nice to be able to do indefinite integration of piecewise functions in Sage.

I've created a patch which does this. I have made the default behavior of the integral() function of a piecewise function be to return the indefinite integral, and the definite integral is returned only when definite=True is supplied.

sage: pw = Piecewise([[(0,1), x*2], [(1,2), x + 3]])
sage: pw.integral()
Piecewise defined function with 2 parts, [[(0, 1), x^2], [(1, 2), (x^2 + 6*x)/2 - 5/2]]
sage: pw.integral(definite=True)
11/2

CC: @mwhansen

Component: calculus

Issue created by migration from https://trac.sagemath.org/ticket/4721

[wdjoyner]

comment:1

I don't agree that this should be the default. First, how is the indefinite integral of piecewise function even well-defined? Assuming it somehow makes sense, the definite integral should be the default and the indefinite one (which should be very well documented) should require options. Also, my vote is that one should require the optional "method=indefinite" as opposed to "definite=True". I think "method" is more of a Sage standard.

[sagetrac-pbutler]

Attachment: piecewise_integration.patch.gz

[sagetrac-pbutler]

comment:2

I made integration the default for consistency with other functions. For example,

sage: f(x)=3
sage: f.integral()
x |--> 3*x
sage: (x*8).integral()
4*x^2
sage: sin(x).integral()
-cos(x)

When integrating a piecewise function in Maple, the result is the indefinite integral. The only difference between the Maple result and the patched Sage code is that Maple defines the integral outside the bounds of the original piecewise function, which my code does not.

[wdjoyner]

comment:3

Okay. It makes good sense to usually copy the default, or implement an idea used in Maple's interface design.

But in this case, the indefinite integral of a piecewise defined function is not well-defined. I have no idea why Maple implemented it. I'm sure they had some reason. But IMHO Sage should not be implementing methods which so not have a well-defined mathematical meaning as a default. As an option (which is very well-documented), I have no objection because whoever would be using it presumably know what it means. However, I don't know what it means to indefinitely integrate a piecewise defined function and your code does not explain it.

It might be useful for some, so could you please revise it a bit?

[sagetrac-pbutler]

comment:4

Interestingly, Mathematica Online Integrator gives an answer too, but chops off the constant in each piece. You still get a valid antiderivative, but it is not useful as an indefinite integral. (Since it is discontinuous, it doesn't have to satisfy the conclusion of the FTC.)

I do see the value of keeping to mathematically well-defined results, but there is also value in consistency. integrate(f) returning an antiderivative in some cases and a definite integral in others doesn't feel intuitive to me.

In any case, I'll clean this code up so that it can be included. It should be useful for people working with continuous probability distributions, if nothing else. (Actually, there must be some other use for it too, because Mathematica does it and their results would not be useful for probability.)

[sagetrac-mabshoff]

comment:5

Hi Paul,

design discussions like this should not be done in trac since the number of people actually getting emails from this is minuscule compared to the audience on sage-devel. So please send an email with a summary what is proposed and what has been discussed so far to sage-devel. Also mention the ticket and that there are some more details there.

Thanks and keep up the good work.

Cheers,

Michael

[sagetrac-pbutler]

Attachment: piecewise_integration2.patch.gz

[sagetrac-pbutler]

comment:6

I've made some changes and added a new patch. Changing from needs work to needs review.