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Riemann, Bernhard

Riemann, Bernhard

Riemann, Bernhard (Georg Friedrich Bernhard Riemann), 1826-66, German mathematician. He studied at the universities of Göttingen and Berlin and was professor at Göttingen from 1859. His great contributions to mathematics include his work on the theory of the functions of complex variables (see complex variable analysis) and his method of representing these functions on coincident planes or sheets (Riemann surfaces). He laid the foundations of a non-Euclidean system of geometry (Riemannian geometry) representing elliptic space and generalized to n dimensions the work of C. F. Gauss in differential geometry, thus creating the basic tools for the mathematical expression of the general theory of relativity. Riemann also was interested in mathematical physics, particularly optics and electromagnetic theory. The Riemann zeta-function analytically encodes information about the distribution of prime numbers. The so called "Riemann hypothesis," concerning the instances in which the function's value is zero, is one of the great unsolved problems in mathematics.

See studies by J. Derbyshire (2003), M. du Sautoy (2003), and K. Sabbagh (2003).

(born Sept. 17, 1826, Breselenz, Hanover—died July 20, 1866, Selasca, Italy) German mathematician. He studied at the Universities of Berlin and Göttingen and later taught principally at Göttingen. His dissertation (1851) was on function theory. He became convinced that mathematical theory could link magnetism, light, gravitation, and electricity and suggested field theories, in which the space surrounding electrical charges may be mathematically described. While continuing to develop unifying mathematical themes in the laws of physics, he created Riemannian geometry (or elliptic geometry), which proved essential to Albert Einstein's model of space-time in relativity theory. Riemann surfaces, Riemann integrals, and Riemann curvature, among other concepts, contributed to the understanding of curves and surfaces, as well as of calculus. With Carl Friedrich Gauss, Riemann helped establish Göttingen's reputation as a world leader in mathematical research. His work widely influenced geometry and analysis.

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In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.

Definition

The Riemann-Stieltjes integral of a real-valued function f of a real variable with respect to a real function g is denoted by

int_a^b f(x) , dg(x)

and defined to be the limit, as the mesh of the partition P of the interval [a, b] approaches zero, of the approximating sum

sum_{x_iin P}f(c_i)(g(x_{i+1})-g(x_i))

where ci is in the i-th subinterval [xi, xi+1]. The two functions f and g are respectively called the integrand and the integrator. Most commonly, g will be nondecreasing, but this is not required. In order that this Riemann-Stieltjes integral exist it is necessary that f and g do not share any points of discontinuity.

An alternative, and slightly more general, definition of the Riemann-Stieltjes integral uses the same approximating sums as above, but takes the limit to be a Moore-Smith limit on the directed set of partitions of [a, b]. That is, take the limit as more and more division points are inserted into the partition. With this definition, an integral can exist when f and g share points of discontinuity, as long as they are not discontinuous from the same side at the same point.

For another formulation of integration that is much more general, see Lebesgue integration. It is notable however, that if improper Riemann-Stieltjes integrals are allowed, the Lebesgue integral is not strictly more general.

Properties and relation to the Riemann integral

If g should happen to be everywhere differentiable, then the integral may still be different from the Riemann integral

int_a^b f(x) g'(x) , dx,

for example, if the derivative is unbounded. But if the derivative is continuous, they will be the same. This condition is also satisfied if g is the (Lebesgue) integral of its derivative; in this case g is said to be absolutely continuous.

However, g may have jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, g could be the Cantor function or Minkowski's question mark function), in either of which cases the Riemann-Stieltjes integral is not captured by any expression involving derivatives of g.

The Riemann-Stieltjes integral admits integration by parts in the form

int_a^b f(x) , dg(x)=f(b)g(b)-f(a)g(a)-int_a^b g(x) , df(x).

and the existence of the integral on the left implies the existence of the integral on the right.

Existence of the integral

The best simple existence theorem states that if f is continuous and g is of bounded variation on [a, b], then the integral exists. Note that g is of bounded variation if and only if it is the difference between two monotone functions. If g is not of bounded variation, then there will be continuous functions which cannot be integrated with respect to g.

Application to probability theory

If g is the cumulative probability distribution function of a random variable X that has a probability density function with respect to Lebesgue measure, and f is any function for which the expected value E(|f(X)|) is finite, then, as is well-known to students of probability theory, the probability density function of X is the derivative of g and we have

E(f(X))=int_{-infty}^infty f(x)g'(x), dx.

But this formula does not work if X does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of X is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function g is continuous, it does not work if g fails to be absolutely continuous (again, the Cantor function may serve as an example of this failure). But the identity

E(f(X))=int_{-infty}^infty f(x), dg(x)

holds if g is any cumulative probability distribution function on the real line, no matter how ill-behaved.

Application to functional analysis

The Riemann-Stieltjes integral appears in the original formulation of F. Riesz theorem which represents the dual space of the Banach space C[a,b] of continuous functions in an interval [a,b] as Riemann-Stieltjes integrals against functions of bounded variation (later, that theorem was reformulated in terms of measures).

Also, the Riemann-Stieltjes integral appears in the formulation of the spectral theorem for (non-compact) self-adjoint (or more generally, normal) operators in a Hilbert space (in this theorem, the integral is considered with respect to a so-called spectral family of projections).

[see the book by F. Riesz for details]

See also

References

  • Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.
  • Stroock, Daniel W., 1998. A Concise Introduction to the Theory of Integration. Birkhauser. 3 edition. ISBN 0-8176-4073-8. Includes problems with solutions.
  • F. Riesz, B. Sz. Nagy. Functional Analysis. (1955) F. Ungar Publishing.

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