3Blue1Brown
The Borwein integrals, a sequence of integrals that appear to be unrelated, are shown to be connected through Fourier transforms. The video explains the use of stretched and multiplied versions of the sinc function to demonstrate how integrals generate a stable area equal to pi until they fall just short of ideal value, with a similar factor in front of pi. By evaluating Fourier transforms of functions, the video shows how Borwein integrals can be computed based on the Fourier transform of a function, and how convolutions can be understood using moving averages. Overall, the video highlights the power of Fourier transforms and the convolution theorem to solve tricky problems.
In this section, we explore a sequence of computations revolving around the main character, the function sine of x divided by x, commonly known as "sinc." The integral of this function from negative infinity to infinity equals pi, which is strange and not easily calculated using conventional calculus. The sequence progresses by taking stretched versions of the sinc function, multiplying them, and finding the integral, with each iteration equalling pi and progressively squishing down. It's peculiar that the area remains stable for so long until it breaks at 15 in the tiniest way possible. Additionally, if all the integrals include an extra factor 2cos(x), the same pattern emerges until falters at 113 by a minuscule amount. This phenomenon was described by Jonathan and David Borwein, and there is a satisfying explanation for it.
In this section, the speaker defines a sequence of functions that are moving averages of the previous function with varying window widths. As the process is repeated, the plateau in the output function gets thinner and thinner. This process is analogous to the Borwein integrals where a stable output is obtained until it falls just short of the ideal value, with a similar factor that sits in front of pi in the series of integrals. The addition of 2 cosine of X inside the integral causes the pattern to last longer before it breaks down and corresponds to a function with a longer plateau. The computation involves adding reciprocals of odd numbers until the sum becomes bigger than two, which occurs after 113 iterations.
In this section, the video explains how Borwein Integrals - a family of seemingly unrelated integrals - are actually connected through Fourier Transforms, using a normalized function called the "sinc" function. By replacing X with π*x, the integral patterns seem to continue indefinitely until 113, but as soon as the sum of those numbers is greater than one, the expression drops below pi. The video explains what Fourier Transforms are and how they transform a function into a new language, making it easier to answer questions about that function. The video also explains how evaluating the Fourier Transformed version of a function at input zero is the same thing as computing the integral of that function from negative infinity to infinity. By knowing that the sinc and rect functions are related through the Fourier Transform, the video shows how the seemingly unrelated Borwein Integrals can be computed by finding the Fourier Transform of a function.
In this section, the video discusses the connection between taking Fourier Transforms and convolutions, specifically in the case of rectangular functions. It is explained that taking a convolution looks like a moving average, combined with the fact that integrating looks like evaluating at zero, and that multiplying corresponds to these progressive moving averages, it explains why multiplying more and more sinc functions together can be thought of in terms of these moving averages and always evaluating at zero. This understanding provides a glimpse of why Fourier Transforms are powerful tools and can help solve tricky problems. The video also teases that the convolution theorem opens doors for an algorithm that lets you compute the product of two large numbers faster than you think is possible.
No videos found.
No related videos found.
No music found.