Researchers thought this was a bug (Borwein integrals) [video]

https://www.youtube.com/watch?v=851U557j6HE

by andersource • 4 years ago

110 18 4 years ago

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selimthegrim2• 4 years ago

I’m waiting for the convolution video he promises in the video and the comments.

selimthegrim2• 4 years ago

It’s posted: https://youtu.be/KuXjwB4LzSA

wallfacer120• 4 years ago

Grant Sanderson redeems the entire Internet.

cvoss• 4 years ago

For those who'd like to read a quick description of what this video is about:

It shows a sequence of integrals following a very simple pattern. The first seven integrals in the sequence all evaluate to pi. The eighth integral inexplicably evaluates to pi - 0.0000000000462... and from that point on the pattern deviates from pi.

The video goes on to explain how such a seemingly perfect pattern can suddenly break by relating this sequence of integrands to a simpler one where it's easier to see what's happening.

krackers3• 4 years ago

One-line summary is to look at the fourier transforms of the integrand, fhat(x), then the integral is just fhat(0). The pattern is to multiply the integrand by sinc(x/k) for increasing k, and by fourier transform this is krect(komega). Since multiplication in time-domain is convolution in fourier domain, convolving rect with itself keeps eroding the edges of the rect, until eventually the erosion reaches the center and fhat*(0) dips.

greesil• 4 years ago

Don't you mean convolution in the time domain is multiplication in the "Fourier" domain?

acchow2• 4 years ago

I’m shocked how succinctly you were able to summarize the ideas here. Bravo. Any chance you have a blog?

Gordonjcp• 4 years ago

I understood electronic differentiators and integrators long before we did calculus in high school and I've always had more of a practical than mathematical leaning with that sort of stuff.

I guess what you're saying here is that if I pass a square wave through a lowpass filter - reducing the amplitude of the harmonics and rounding off the corners - then the peak amplitude will stay pretty much constant until I pull the cutoff of the filter down sufficiently close to the fundamental that it starts getting attenuated too.

Makes sense I guess.

acchow2• 4 years ago

I love all the videos on this channel. The partial differential equations tour is incredible too.

gugagore3• 4 years ago

I'd like to know what function this converges to:

rect(x) • rect(x/2) • rect(x/4) • ...

Where • is the convolution operator.

Unlike the series in the video, 1 + 1/2 + 1/4 + ... converges. So this function has compact support, and the value at 0 does not dip.

I expect it to be a https://en.m.wikipedia.org/wiki/Bump_function

eutectic• 4 years ago

I guess it's a Gaussian, by the central limit theorem.

gugagore3• 4 years ago

No, it can't be. This function has compact support, the Gaussian clearly does not.

You get a Gaussian by repeated convolution of the same function (and normalizing the width).

The equivalent question to what I asked is what's the pdf of X1 + X2/2 + X3/4 + ...

where Xi is iid unit uniform.

krackers3• 4 years ago

Do you mean `rect(2x) • 2rect(2x) • 4rect(4x)` .., both so that the limiting function is a dirac delta and the convolutions remain area preserving? Otherwise since the function you're convolving with keeps getting wider wouldn't the result also keep getting wider, so the support is no longer finite?

It might be possible to get a closed-form solution via an approach like [1]. (Out of curiosity, for rect convolved with itself, the intermediaries seem to be knwon as b-splines: https://www.chebfun.org/examples/approx/BSplineConv.html)

[1] https://math.stackexchange.com/questions/1254392/the-maximum...

gugagore3• 4 years ago

Yes that is what I meant, thanks!

bheadmaster• 4 years ago

I love this.

I sometimes regret not studying pure math in college, and going down the software engineer (ahem, code monkey) route. There's so much mathematical beauty out there to be discovered and admired.

But I guess money's better this way.

jojobas• 4 years ago

I went with applied maths route that had a lot of pure math, yet I'm a code monkey anyway. There is beauty in maths but while I was able to graduate I'm too dumb for advancing it, and I'm ok with that.

Researchers thought this was a bug (Borwein integrals) [video]

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