Researchers thought this was a bug (Borwein integrals)
3Blue1Brown・19 minutes read
The text explores the stable results of the sinc function and its integral equating to pi, despite varying manipulations and iterations. It delves into the relationship between the sinc function, moving averages, Fourier transforms, and convolutions, highlighting stable values and deviations from pi.
Insights
- The sinc function, when integrated from negative infinity to infinity, consistently yields a value of pi, showcasing a surprising and stable mathematical result despite variations in function manipulation.
- The connection between the integral sequence of the sinc function and the moving average sequence is elucidated through Fourier transforms and convolutions, unveiling a profound relationship between seemingly disparate mathematical phenomena and shedding light on the underlying patterns of stability and slight deviations encountered in both sequences.
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Recent questions
What is the significance of the sinc function in math and engineering?
The sinc function, sine of x divided by x, is a crucial mathematical tool used in various fields like math and engineering. It plays a key role in signal processing, Fourier transforms, and filtering due to its unique properties and stable results.
How does multiplying stretched versions of the sinc function result in complex waves?
Multiplying stretched versions of the sinc function generates complex waves, yet the signed area underneath remains constant at pi. This phenomenon showcases the stability of the sinc function despite changes in its shape, leading to intricate wave patterns.
What is the relationship between the integral sequence and the moving average sequence?
The integral sequence and the moving average sequence exhibit similar patterns of stable values breaking slightly at specific points. The connection between the two sequences is explained through Fourier transforms and convolutions, revealing a deeper relationship between seemingly unrelated phenomena in mathematics.
How does adding a factor like 2 cosine of x impact the stability of the moving average sequence?
Adding a factor like 2 cosine of x extends the stability of the moving average sequence before breaking, resulting in longer plateaus. This factor influences the behavior of the sequence, showcasing how small modifications can affect the overall stability of mathematical functions.
What is the role of Fourier transforms in understanding the relationship between sinc and rect functions?
Fourier transforms play a crucial role in rephrasing functions like the sinc and rect functions in a new language, showcasing their connection and underlying patterns. By breaking down functions into pure frequencies, Fourier transforms provide insights into the relationship between different mathematical functions and their properties.
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Summary
00:00
"Stable Pi Patterns in Sinc Functions"
- The sequence of computations follows a predictable pattern, each equaling pi, but eventually stops slightly below pi.
- The main focus is on the function sine of x divided by x, commonly known as sinc, which is used in math and engineering.
- The integral of the sinc function from negative infinity to infinity equals pi, a surprising and stable result.
- Multiplying the sinc function by stretched versions results in complex waves, yet the signed area underneath still equals pi.
- Iterating with increasingly stretched versions of the sinc function continues to yield an area of pi, despite expectations of change.
- The pattern of stable results breaking slightly occurs at specific values, such as 15 and 113, with minimal deviations from pi.
- A similar phenomenon is observed in a sequence of functions defined by moving averages, where stable values eventually break slightly.
- The breaking point in the moving average sequence aligns with the breaking points in the integral sequence, both deviating slightly from stability.
- Adding a factor like 2 cosine of x extends the stability before breaking, corresponding to longer plateaus in the moving average sequence.
- The connection between the integral sequence and the moving average sequence is explained through Fourier transforms and convolutions, revealing a deeper relationship between the two seemingly unrelated phenomena.
11:33
"Pi Sinc Function and Fourier Transform"
- Replacing x with pi times x squishes the graph horizontally by a factor of pi, scaling down the area by the same factor.
- The integral on the right needs to be proven to be equal to 1, with the function sometimes referred to as sinc in engineering contexts.
- The sinc function with pi inside is related to the rect function through a Fourier transform, which breaks down functions into pure frequencies.
- The Fourier transform rephrases functions in a new language, showing the connection between the sinc and rect functions.
- Multiplying sinc functions corresponds to convolutions, akin to moving averages, explaining the stable value until the edges of the plateau approach the center.
