The term usually denotes a limiting process which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when
...[a,b[$. If the limit exists and is finite, then one says that the improper integral converges and if not, that it diverges. A similar definition is possible fo
7 KB (1,076 words) - 12:30, 7 July 2013
is a polynomial, called the integral part of the fraction $ P / Q $,
in (1), can be found by the method of indefinite coefficients (cf. [[Undetermined coefficients, method of|Undetermined coeff
8 KB (1,257 words) - 03:49, 4 March 2022
...1 = 89 : ~/encyclopedia/old_files/data/K110/K.1100200 Kurzweil\ANDHenstock integral
...[[#References|[a10]]], gave an integral that is equivalent to the Lebesgue integral.
11 KB (1,718 words) - 22:15, 5 June 2020
and if the indefinite integral of $ f ( s) $
3 KB (422 words) - 19:35, 5 June 2020
are integers, the binary quadratic form is said to be integral. The expression $ d = ac - {b ^ {2} } /4 $
All integral automorphisms of a primitive form $ f $
7 KB (977 words) - 15:17, 31 March 2024
...ion, partial]]); the theory of integral equations (cf. [[Integral equation|Integral equation]]); [[Differential geometry|differential geometry]]; [[Variational
...es|Fourier series]], and Fourier integrals (cf. [[Fourier integral|Fourier integral]]).
19 KB (2,844 words) - 08:57, 13 January 2024
...by series (in particular, by trigonometric series), the primitive and the integral, relations between integration and differentiation, term-by-term integratio
...e integral|Lebesgue integral]]). These two fundamental ideas — measure and integral — formed the foundation of the metric theory of functions, which is conce
11 KB (1,633 words) - 08:00, 6 June 2020
a primitive of $u$, the integration by parts formula for the definite integral $\int_a^b
...The latter assertion is also called ''formula of integration by parts for indefinite integrals''.
3 KB (493 words) - 09:25, 16 August 2013
is called the indefinite inner product of the Krein space $ {\mathcal K} $.
then an indefinite inner product is defined on $ {\mathcal K} $
32 KB (4,628 words) - 10:55, 20 January 2024
...hinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin).
|valign="top"|{{Ref|Sa}}|| S. Saks, "Theory of the integral" , Hafner (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
5 KB (745 words) - 17:18, 18 August 2012
...uences of algebraic operations plus the operation of taking the indefinite integral (quadratures) of the functions given, then one speaks of the integration (i
4 KB (607 words) - 15:57, 14 February 2020
The indefinite integral $ F( x)= \int _ {0} ^ {x} f( t) dt $
5 KB (793 words) - 08:05, 6 June 2020
is an $ ( n + p ) $-dimensional indefinite [[Riemannian manifold|Riemannian manifold]] of index $ p $
...D></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Montiel, "An integral inequality for compact space-like hypersurfaces in a de Sitter space and ap
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(an analogue of the second metric of example 2)), and the integral metric
...ns which do not have all the properties 1)–3); such are, for example, an [[indefinite metric]], a [[symmetry on a set]], etc.
5 KB (793 words) - 09:16, 7 June 2016
...tions is larger than the number of equations; thus, they are also known as indefinite equations. In modern mathematics the concept of a Diophantine equation is a
...the scientists of Ancient Greece led to the study of rational solutions of indefinite equations. This point of view is systematically followed by Diophantus in h
18 KB (2,685 words) - 08:57, 18 August 2022
...and quotient representations, of decomposing representations into a direct integral, and of studying dual objects, and the problems of the theory of spherical
...an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]). A complete description of the irreducible representations in suc
20 KB (2,775 words) - 16:40, 31 March 2020
itself either negative definite or indefinite, then the equilibrium is unstable.
...e to a system of equations with constant coefficients and have a quadratic integral of definite sign.
7 KB (1,104 words) - 13:23, 6 January 2024
...1 }$, which play an important role in the study of spectral properties of integral operators of Carleman type (cf. also [[Carleman operator|Carleman operator]
...d to both finite Hankel matrices, by considering discrete measures, and to indefinite Hankel matrices, by means of formal orthogonal polynomials. A survey of res
14 KB (2,126 words) - 16:45, 1 July 2020
...ed under a discrete change of the argument, as opposed to differential and integral calculus, where the argument changes continuously. Let $ f $
27 KB (3,799 words) - 17:14, 12 November 2023
...ightman functions based on the classical Lagrangian and the heuristic path integral (cf. also [[Quantum field theory|Quantum field theory]]).
...]]]; for models with $d$ arbitrary (however with a state space carrying an indefinite inner product), see e.g. [[#References|[a12]]].
9 KB (1,385 words) - 20:57, 8 February 2024