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The y-intercept of the parabola is − + 1 / 12 . [1] The method of regularization using a cutoff function can "smooth" the series to arrive at − + 1 / 12 . Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler ...
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. ′ = which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of J [ f ] {\displaystyle J[f]} and is denoted δ J {\displaystyle \delta J} or δ f ( x ...
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method assigns = to / for all in a subset of the complex plane, given certain restrictions on , then the method also gives the analytic continuation of any other function () = = on ...
In mathematical analysis, a summability method is an alternative formulation of convergence of a series which is divergent in the conventional sense. Subcategories This category has the following 2 subcategories, out of 2 total.
Euler–Boole summation is a method for summing alternating series. The concept is named after Leonhard Euler and George Boole . Boole published this summation method, using Euler's polynomials , but the method itself was likely already known to Euler.
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus .
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]