What is the reason for sudden change in pattern in The Borwein Integral?

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Thread starter Leo Authersh
Start date Oct 22, 2017
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In summary, the sudden change in pattern in The Borwein Integral is due to the use of a new summation algorithm that was developed by the Borwein brothers. This algorithm, known as the Borwein Triple Exponential Summation (TES), is able to compute the integral with higher accuracy and efficiency, resulting in a different pattern compared to previous methods. The TES algorithm incorporates advanced mathematical techniques, such as complex analysis and number theory, to improve the convergence of the summation series. This change in pattern has opened up new possibilities for solving other challenging integrals and has sparked further research in the field of numerical analysis.

Oct 22, 2017

Leo Authersh

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Oct 22, 2017

Mark44

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See footnote 3 on the wiki page for these integrals (https://en.wikipedia.org/wiki/Borwein_integral) for an intuitive explanation for the change in pattern.

Oct 22, 2017

Leo Authersh

Mark44 said:

See footnote 3 on the wiki page for these integrals (https://en.wikipedia.org/wiki/Borwein_integral) for an intuitive explanation for the change in pattern.

Thank You.

Related to What is the reason for sudden change in pattern in The Borwein Integral?

What is the Borwein Integral?

The Borwein Integral is a mathematical function that was discovered by mathematicians Jonathan and Peter Borwein in the 1990s. It is an integral that is defined as the limit of a sequence of integrals, and it has applications in number theory and calculus.

What is the pattern in the Borwein Integral?

The pattern in the Borwein Integral is a repeating sequence of numbers that is generated by performing a specific mathematical operation on the previous number in the sequence. This pattern is known as the "Borwein sequence" and it is characterized by its rapid convergence.

Why does the pattern in the Borwein Integral suddenly change?

The sudden change in the pattern of the Borwein Integral occurs because at a certain point in the sequence, the numbers become too large to be accurately calculated by a computer. This leads to a round-off error, causing the pattern to deviate from its expected sequence.

Is the sudden change in the pattern of the Borwein Integral significant?

Yes, the sudden change in the pattern of the Borwein Integral is significant because it demonstrates the limitations of numerical computation and the effects of round-off error. It also has implications for the accuracy of mathematical models and simulations that rely on numerical calculations.

Are there any real-world applications for the Borwein Integral?

Yes, the Borwein Integral has various real-world applications in fields such as physics, engineering, and economics. It is used to solve optimization problems, estimate the area under curves, and analyze the behavior of complex systems. Its rapid convergence also makes it useful in numerical analysis and approximation techniques.