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Two-sided Laplace Transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral :\mathcal\(s) = F(s) = \int_^\infty e^ f(t)\, dt. The integral is most commonly understood as an improper integral, which converges if and only if both integrals :\int_0^\infty e^ f(t) \, dt,\quad \int_^0 e^ f(t)\, dt exist. There seems to be no generally accepted notation for the two-sided transform; the \mathcal used here recalls "bilateral". The two-sided transform used by some authors is :\mathcal\(s) = s\mathcal\(s) = sF(s) = s \int_^\infty e^ f(t)\, dt. In pure mathematics the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. A convergent series that is not absolutely convergent is called conditionally convergent. Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded in absolute value by an integrable function and is almost everywhere pointwise convergent to a function then the sequence converges in L_1 to its pointwise limit, and in particular the integral of the limit is the limit of the integrals. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Real Number
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence (1,2,\ldots) of the natural numbers increases ''infinitively'' and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has +\infty as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of +\infty and -\infty as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers. The extended real number system is denoted \overline, \infty,+\infty/math>, or \R\cup\left\. When the meaning is clear from context, the symbol +\infty i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single Variable (mathematics), variable, being of bounded variation means that the distance along the Direction (geometry, geography), direction of the y-axis, -axis, neglecting the contribution of motion along x-axis, -axis, traveled by a point (mathematics), point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a Glossary of differential geometry and topology#H, hypersurface in this case), but can be every Intersection (set theory), intersection of the graph itself with a hyperplan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that every point has an open neighborhood with finite measure. For Hausdorff spaces, this implies that \mu(C) 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Integrable
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions. Definition Standard definition .See for example and . Let be an open set in the Euclidean space \mathbb^n and be a Lebesgue measurable function. If on is such that : \int_K , f , \, \mathrmx <+\infty, i.e. its |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of '' almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plancherel Theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science and engineering, proving the unitarity of the Fourier transform. The theorem states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if f(x) is a function on the real line, and \widehat(\xi) is its frequency spectrum, then Formal definition The Fourier transform of an ''L''''1'' function f on the real line \mathbb R is defined as the Lebesgue integral \hat f(\xi) = \int_ f(x)e^dx. If f belongs to both L^1 and L^2, then the Plancherel theorem states that \hat f also belongs to L^2, and the Fourier transform is an isometry with respect to the ''L''2 norm, which is to say that \int_^\infty , f(x), ^2 \, dx = \int_^\infty , \widehat(\xi), ^2 \, d\xi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parseval's Theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
