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2016 HSC
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Advanced Mathematics (Hons)/Computer Science @ UNSW (2017– )
Nvm deleted it. I was mindlessly experimenting with Wolfram when I realised what the f**** I actually did
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Take the Imaginary part of the following integral:
The imaginary component of the integral is absolutely convergent, and so is the Taylor expansion, so (insert reasoning here) it is valid to exchange summation, integration and complex extraction.
Take the Taylor series of e^{x} and replace x with e^{ix}
Swap the order of integration and summation. Integrate termwise.
final answer:
π(e-1)/2
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
The imaginary part of the integrand is the original integrand, which is
This is not absolutely convergent. We have something continuous and periodic divided by x. Upon taking absolute values, integrating this is like summing a harmonic series.
The integral converges in the sense of an improper Riemann integral because it oscillates as well as decays (things like the Dirichlet test or alternating series test pin this notion down), but its rate of decay is too slow to give us absolute convergence.
The most common ways of justifying an interchange rely on our limit function being absolutely integrable (the monotone/dominated/vitali convergence theorems), or on us having an absolutely integrable error term which we can bound and show tends to zero, so this is not as routine as you might think, even if it is probably justifiable somehow.
Last edited by seanieg89; 29 Jun 2016 at 10:11 AM.
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.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
I don't even know at all. When I first saw this integral I had absolutely no idea on what to do obviously.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
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).
Last edited by seanieg89; 29 Jun 2016 at 2:10 PM.
So is this integral from an IB maths textbook?
Still waiting for my source to reply...
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
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