Analysis Fact

@AnalysisFact

Daily tweets about real and complex analysis and related topics. From .

Vrijeme pridruživanja: prosinac 2009.

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  1. prije 2 sata
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  2. prije 3 sata

    The Brachistochrone problem motivated the development of the calculus of variations.

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  3. proslijedio/la je Tweet
    prije 23 sata
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  4. 1. velj

    The harmonic series diverges. Leibniz expressed this as:

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  5. 31. sij

    The Riemann zeta function is log-convex on (1, infinity).

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  6. 31. sij

    The series 1 + z + z^2 + z^3 + ... converges absolutely to 1/(1-z) for |z| < 1. It converges EVERYWHERE to 1/(1-z) with Borel summation.

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  7. 31. sij

    Where you can get into trouble is assuming implicitly that the result of one summation procedure has the properties of the result of another procedure. If you want all the properties of absolute convergence, then you need absolutely convergent series.

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  8. 31. sij

    Summing a divergent series is not sloppy math, it’s sloppy language. It’s using informal language to describe what may be a formally justifiable procedure.

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  9. 31. sij

    The right summation method, if there is one, depends on context. Useful sums converge, but they may converge by new criteria. Asymptotic series converge, just not necessarily by the usual definition. Fix n and let x -> ∞, rather than fixing x and letting n -> ∞.

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  10. 31. sij

    We'd probably like a definition that gives the same answer as the traditional definition if the sum converges. So here's a second attempt: Define the sum to be the classical sum if the classical sum converges, and define it to be 7 otherwise.

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  11. 31. sij

    If you just want an answer, here's one: Define the sum of every series to be 7. This definition has some disadvantages, but at least it's easy to compute.

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  12. 31. sij

    There are many ways to sum a divergent series. Which one is correct? Depends on your criteria. None of them are correct according to the criterion by which the series diverges!

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  13. 31. sij
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  14. 31. sij

    "If you can prove that the candidate solution is correct, then it’s correct, even if you can’t justify the process that led to its discovery."

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  15. 31. sij

    New post: Mittag-Leffler transform and Mittag-Leffler summation

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  16. 30. sij

    The error function is odd. i.e. erf(-x) = -erf(x)

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  17. 30. sij

    A convex function is continuous on the interior of its domain.

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  18. 30. sij

    "One reason for the continuing popularity of special functions could be that they enshrine sets of recognizable and communicable patterns and so constitute a common currency." -- Michael Berry

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  19. 29. sij

    Euler's constant gamma is the limit of 1 + 1/2 + 1/3 +... + 1/n - log(n) as n goes to infinity.

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  20. 29. sij
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