In another forum, someone said to another one to use an analogue of this test for proving this theorem : Differentiation under the integral sign - Wikipedia, the free encyclopedia
I've read that Weierstrass M test is made for series, so is it really possible to use it here ?
Anyway, can anyone give me the guidelines for the proof of this theorem ?
If I've been unclear, please tell me.
I've never read about this analogue of Weierstrass M test for integrals
At least not under this name... So if you could clarify .. ?
The theorem I'm interested in is more or like this one :
Then is (that is continuous and differentiable) and we have :
And it's for proving this stuff that someone suggested to use the Weierstrass M test
Let's begin with the "elementary" one. Before differentiation, let's deal with continuity since it is more natural to begin with.
Proposition: Suppose is continuous, where is an open interval (or a metric space, for instance). Then is continuous on .
Proof: Let . Let . Choose a compact neighbourhood of contained in (like for some small ). The function is continuous on , which is compact, hence is uniformly continuous on . In particular, there is such that, if (and ), then for every , . As a consequence, for these values of , . This is it.
Proposition: Suppose is differentiable with respect to the first variable and that is continuous on . Then is differentiable on , and .
Proof: Let . In the same way as above, we can find a compact containing , and we say that the function is uniformly continuous on , hence there is such that, if , then . Then, the conclusion comes from:
These proofs are nice because of their simplicity, but they require integration on a compact space to get uniform continuity. For the integration on a more general interval, an additional hypothesis (a "domination") is needed. This is where Lebesgue's dominated convergence theorem comes into play.
Lebesgue's dominated convergence theorem is a bit like what MathStud28 wrote, but his statement is not correct (why would the sequence of integrals converge?). The dominated convergence theorem is a theorem of "limit under the integration sign". It says: (you can replace "measurable" by "piecewise continuous", or almost whatever that makes the integration make sense for you)
Theorem: suppose , , and are (measurable) functions (where is an interval), such that for all (or almost all), , for every and every , , and such that ( is integrable on ). Then, for every , is integrable on , and .
I won't prove it because it is definitely not an "elementary" theorem; it is not even clear why can be integrated (a limit of piecewise continuous functions doesn't have to be piecewise continuous). But it is simple to deduce the continuity and differentiation theorems from this one. The bounded convergence theorem deals with sequences, so we shall use sequences instead of , which is equivalent (if we consider any sequence converging to 0).
Proposition: if is continuous with respect to the first variable (and measurable), and if there is a (measurable) function such that for any and , then is continuous on .
Proof: Notice is well-defined thanks to the domination by . let . Let be a sequence converging to 0. Then , and we apply the dominated convergence theorem to : we have , and , hence . This is it.
Proposition: Let be differentiable with respect to the first variable, and such that there is with for all , and . Then is well-defined, differentiable on , and .
Proof: To see that is well-defined, you can notice that you can deduce an upper bound for from the upper bound given by assumption, and this upper bound does not depend on when is in a compact. Never mind, because it is usually easy to see that is well defined, by other means. Now, to show that it is differentiable at , let be a sequence converging to 0. Let us write , where , and apply the bounded convergence theorem. We have , and the domination is a consequence of the mean value theorem: , for some , so that . Hence . And this is it.
I hope you enjoyed it... "The dominated convergence theorem" translates as "le théorème de convergence dominée" and this is a must-know in higher studies.
About the case when the bounds of the integral depend on ...
I assume here a little bit of knowledge about functions of several variables, namely that if the partial derivatives of a function exist and are continuous at one point, then the function is differentiable at that point.
Let us consider , where is continuously differentiable with respect to and continuous with respect to .
We can write where . The interest in doing this is that we know how to differentiate with respect to any of its three variables: (this is what we did above), and (this is the usual property relating primitive function and integration).
These three partial derivatives are continuous (for the first one, because of the continuity theorem, and a bit more). Hence the function is differentiable. So that, by chain rule,
and this should give the formula on the Wikipedia.
Hello Moo, I am going to be different from Laurent and I hope this is okay for you. Tell me if you do not like it and maybe we can find something better. I tell you this because this is not as general as you want
Theorem 1: Let be a rectangle in and (if you perfer ) be a continous function on with a continous function on . Then we have that, for all .
In order to prove this we use a theorem called Dominated Convergence Theorem.
Theorem 2: For ever real number in a interval around let be a piecewise continous function on a finite interval (a function of - to be a little informal here but hopefully clearer). Say that defines a piecewise continous function, call it . If there is a piecewise continous function such that on then .
Theorem 2 is basically passing the limit under the integral sign. We will not prove this result. If you wish I can try to post a proof of this but accept it for now. With this we can prove Theorem 1.
Proof: For a particular we have by definition:
Let be a maximum for on .
Then we see that,
Thus, we see that for near that .
Define, for and (remember is fixed) for each .
Let on .
We see that by the above work.
Therefore, by Theorem 2 we see that,
Thank you very much to you two !
I'm gonna print these messages and read them again...I have some difficulties to follow such things on a screen...
Yes, it is very clear, and yup, I know this theorem. I was more interested in proofs and hence my questions ^^I hope you enjoyed it... "The dominated convergence theorem" translates as "le théorème de convergence dominée" and this is a must-know in higher studies.
It looks like a lecture !
it is possible that there are some unclear points, if so, I'll give it a feedback asap.
As soon as it uses the Dominated convergence theorem, it is ok, it's these types of proofs I was looking forHello Moo, I am going to be different from Laurent and I hope this is okay for you. Tell me if you do not like it and maybe we can find something better. I tell you this because this is not as general as you want
Once again, thanks a bunch guys !
- When you consider integration on a segment , the constant functions are integrable and thus can be used as a domination (i.e. as the function in my post, or in ThePerfectHacker's). That's why the continuity of the derivative is sufficient to prove the domination in this case: a continuous function on a segment is bounded. In this setting, as I said, I however prefer to use uniform continuity (my first proof) because this is easier, but both approaches are valid of course.
- To avoid using limits along a sequence like me for the continuity and the differentiation, ThePerfectHacker included it once and for all in his version of the dominated convergence theorem. This simplifies the redaction indeed.