Higher Level Integration Marathon & Questions (1 Viewer)

[InteGrand]

Re: Extracurricular Integration Marathon

So is this integral from an IB maths textbook?

 

[Paradoxica]

Re: Extracurricular Integration Marathon

So is this integral from an IB maths textbook?

According to Leehuan

 

[InteGrand]

Re: Extracurricular Integration Marathon

According to Leehuan

What techniques of integration do they learn in IB? Do they learn Complex Analysis methods? I'm guessing they'd do double integrals?

 

[leehuan]

Re: Extracurricular Integration Marathon

Still waiting for my source to reply...

 

[Paradoxica]

Re: Extracurricular Integration Marathon

We are in the extracurricular marathon, so post a contour integration solution if you do find one.

The slow decay of the integrand again makes things nontrivial (what contour do you suggest?), and the singularity is removable, not a pole so there is no benefit in contouring about it (for the original function, it is a pole of the complexified function, but the problem of slow decay remains).

The classical semicircular contour indented at the origin works for every power series term (Jordan's Lemma) but the problem still remains at bay: How do we justify interchange of summation and integration?

 

[leehuan]

Re: Extracurricular Integration Marathon

Haven't gotten a proper answer but right now he said some challenge question from "Quizlet"

 

[InteGrand]

Re: Extracurricular Integration Marathon

Haven't gotten a proper answer but right now he said some challenge question from "Quizlet"

Is Quizlet an online thing, or paper-based?

 

[leehuan]

Re: Extracurricular Integration Marathon

Is Quizlet an online thing, or paper-based?

Turns out it was an...app

 

[InteGrand]

Re: Extracurricular Integration Marathon

Turns out it was an...app

Nice. So it's not actually IB-related?

 

[leehuan]

Re: Extracurricular Integration Marathon

Nice. So it's not actually IB-related?

My assumption is that it was tagged as an IB challenge question. (That's what he implied at least)

 

[Paradoxica]

Re: Extracurricular Integration Marathon

Without invoking the Dilogarithm, or using infinite sums of power series, evaluate

 

[Mongoose528]

Re: Extracurricular Integration Marathon

Is this a year 12 topic?

 

[Paradoxica]

Re: Extracurricular Integration Marathon

Is this a year 12 topic?

No, the Dilogarithm is beyond the curriculum, as is practically everything else on this thread.

 

[leehuan]

Re: Extracurricular Integration Marathon

Is this a year 12 topic?

Integration is a topic in Year 12, however the extent to which it goes is not massive. Even the MX2 Integration Marathon tends to feature questions that are beyond what's examinable in the HSC, let alone the stuff here.

 

[Paradoxica]

Re: Extracurricular Integration Marathon

Here's an easier one. Fun Exercise

 

[Paradoxica]

Extracurricular Integration Marathon

There has to be a less potentially dodgy way of doing it, I also thought of expanding out a series in e^(ix) to start with, but I couldn't see any way of justifying that so I left it.

Gotta be careful with these things, because functions/sequences that oscillate and decay but are not absolutely integrable are a common source of counterexamples to otherwise believable claims.

Fourier expansion. Is that less dodgy, or more?

Also something about the Zeta Regularisation technique.

 

[seanieg89]

Re: Extracurricular Integration Marathon

Fourier expansion. Is that less dodgy, or more?

Also something about the Zeta Regularisation technique.

Well a power series in e^(ix) IS a Fourier series. The difficulty is in getting a mode of convergence that allows us to commute the limit with the conditionally convergent integral. I did think of an idea that probably deals with it though, and will post it during an afternoon break today if it works.

 

[Paradoxica]

Re: Extracurricular Integration Marathon

Find this limit:



Hence, evaluate this:

 

[Paradoxica]

Re: Extracurricular Integration Marathon

A fairly easy exercise.

 

[Paradoxica]

Re: Extracurricular Integration Marathon

Well a power series in e^(ix) IS a Fourier series. The difficulty is in getting a mode of convergence that allows us to commute the limit with the conditionally convergent integral. I did think of an idea that probably deals with it though, and will post it during an afternoon break today if it works.

Well the thing I was looking at for Zeta Regularisation was to use the method I used back in the early days of this thread

http://community.boredofstudies.org...ricular-integration-marathon.html#post7092858

But the thing is the justification...