Riemann in a sentence

A restatement of the Riemann hypothesis The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli number.

For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface.

Integration Riemann integration main The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.

However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral.

In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.

Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations.

Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the analytically extended Riemann zeta function of a complex variable.

Riemann's flawed proof depended on the Dirichlet principle (which was named by Riemann himself), which was considered sound at the time.

Smooth Riemann mapping theorem In the case of a simply connected bounded domain with smooth boundary, the Riemann mapping function and all its derivatives extend by continuity to the closure of the domain.

The imaginary company had produced a proof of the Riemann Hypothesis but then had great difficulties collecting royalties from mathematicians who had proved results assuming the Riemann Hypothesis.

The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough. citation One important requirement is that the mesh of the partitions must become smaller and smaller, so that in the limit, it is zero.

The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function ).

Riemann Labs did not respond to requests for comment from Reuters.

A better approach replaces the rectangles used in a Riemann sum with trapezoids.

Also, both curves are rational, as they are parameterized by x, and Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.

Although this attempt failed, it did result in Riemann finally being granted a regular salary.

Among Rolf Nevanlinna's later interests in mathematics were the theory of Riemann surfaces (the monograph Uniformisierung in 1953) and functional analysis (Absolute analysis in 1959, written in collaboration with his brother Frithiof).

Any complex nonsingular algebraic curve viewed as a complex manifold is a Riemann surface.

A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant.