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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
Georg Friedrich Bernhard Riemann (German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] ⓘ; [1] [2] 17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry.
Numberphile is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. [2] [3] In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, [4] featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis [5] and Kruskal's tree ...
Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem. [31] [32] [33] A historically important example of conditional convergence is the alternating harmonic series,
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
Riemann mapping theorem. Measurable Riemann mapping theorem; Riemann problem; Riemann solver; Riemann sphere; Riemann–Hilbert correspondence; Riemann–Hilbert problem; Riemann–Lebesgue lemma; Riemann–Liouville integral; Riemann–Roch theorem. Arithmetic Riemann–Roch theorem; Riemann–Roch theorem for smooth manifolds; Riemann–Roch ...
In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month. [31] The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham.
In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function.Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. [1]