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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 .
Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory. He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behaviour of closed paths about singularities (described by the monodromy matrix ).
R. Removable singularity; Riemann (crater) Riemann curvature tensor; Riemann form; Riemann hypothesis; Riemann integral; Riemann invariant; Riemann mapping theorem
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.
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]
The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζ K (s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2. The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q , with ring of integers Z .
The function has the series expansion = = +, where = ()! [ ()] | = = [()], where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of | |.. This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λ n > 0 for all positive n.
A function F(x) is an h-antiderivative of f(x) if D h F(x) = f(x).The h-integral is denoted by ().If a and b differ by an integer multiple of h then the definite integral () is given by a Riemann sum of f(x) on the interval [a, b], partitioned into sub-intervals of equal width h.