Interchanging summation with integral, differentiation with integral

[SadScholar]

Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two because I am a physicist and I am lazy. A mathematician would spend his whole life trying to prove this is permissible."

The same goes for "differentiating under the integral," which is what I'm really concerned about. I know that there are times when it's perfectly acceptable to slip that partial differentiation right in under the integral, but I've also come across integrals where it's absolutely not permitted, and it gives you wonky, nonsensical results.

So here is my question. Does anyone have any tricks or rules of thumb, maybe not for always knowing when these things are allowed, but knowing when they are unquestionably allowed. Is there ever a time you can look at such a thing, and say, "OK, I can absolutely interchange these without negative consequence?" I don't want anything too formally mathematical. Just any of your intuitive sense on the topic would be greatly appreciated.

 

[pwsnafu]

Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two because I am a physicist and I am lazy. A mathematician would spend his whole life trying to prove this is permissible."

No mathematician spends their whole life trying to prove that. Proper mathematicians are lazy. And just point to the Dominated Convergence Theorem. Or one of its corollaries.

The same goes for "differentiating under the integral," which is what I'm really concerned about. I know that there are times when it's perfectly acceptable to slip that partial differentiation right in under the integral, but I've also come across integrals where it's absolutely not permitted, and it gives you wonky, nonsensical results.

Differentiation under integral sign

 

[micromass]

What you actually want to switch a sum and an integral is the Fubini theorem for general measures.

Generally, you can do

[tex]\int{\sum_n{f_n}}=\sum_n{\int{f_n}}[/tex]

if either

1) each [itex]f_n\geq 0[/itex]

or if

2) [itex]\int{\sum_n{|f_n|}}<+\infty[/itex] (which by (1) is equivalent to [itex]\sum_n{\int{|f_n|}}<+\infty[/itex])

 

[Castilla]

Let S be the integral sign. Suppose you have Sf(x,y)dy, with constant limits of integration.

You want to justify d/dx S f(x,y)dy = S d/dx f(x,y)dy.

If the partial derivative w.r.t. x of f(x,y) is continuous in a compact set, that is enough justification,
though maybe not necesary. That might be your "rule of thumb".

 

[SadScholar]

Yes! Thankyou guys. That's exactly the kind of stuff I was looking for. Castilla, when you say "constant limits of integration," would an upper bound of infinity be considered such a thing?

 

[Castilla]

I was referring to fixed numbers. Differentiating under the integral sign is also valid with improper integrals ("infinity" in the limit of integration) but in that case you need more requisites. Uniform convergence, I believe.

 

[micromass]

Here is a general criterion for interchanging integral and derivative:

Let [itex]X[/itex] be any interval (or generally: any measurable space). Suppose that [itex]f:X\times [a,b]\rightarrow \mathbb{R}[/itex] and that [itex]f(\cdot,t):X\rightarrow\mathbb{R}[/itex] is integrable for each [itex]t\in [a,b][/itex]. Suppose that [itex]\frac{\partial f}{\partial t}[/itex] exists and there is a pointswize continuous function (or generally: measurable function) [itex]g:X\rightarrow \mathbb{R}[/itex] such that

[tex]\int_X|g(x)|<+\infty~\text{and} ~\left|\frac{\partial f}{\partial t}(x,t)\right|\leq g(x)[/tex]

for all x and t. Then

[tex]\frac{d}{dt}\int_X{f(x,t)dx}=\int_X{\frac{\partial f}{\partial t}(x,t)dx}[/tex]