Search results
Results from the WOW.Com Content Network
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 .
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.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
Riemann form; Riemann hypothesis; Riemann integral; Riemann invariant; Riemann mapping theorem; Riemann problem; Riemann series theorem; Riemann solver; Riemann sphere; Riemann sum; Riemann surface; Riemann xi function; Riemann zeta function; Riemann–Hilbert correspondence; Riemann–Hilbert problem; Riemann–Lebesgue lemma; Riemann ...
When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) Darboux sum. A function is Darboux integrable if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the ...
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Riemann–Roch theorem. Arithmetic Riemann–Roch theorem; Riemann–Roch theorem for smooth manifolds; Riemann–Roch theorem for surfaces; Grothendieck–Hirzebruch–Riemann–Roch theorem; Hirzebruch–Riemann–Roch theorem; Riemann–Stieltjes integral; Riemann series theorem; Riemann sum
Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below). More advanced methods are required, such as zeta function regularization or Ramanujan summation.