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Fractional Calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D D f(x) = \frac f(x)\,, and of the integration operator J The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar glyphs, such as identities. J f(x) = \int_0^x f(s) \,ds\,, and developing a calculus for such operators generalizing the classical one. In this context, the term ''powers'' refers to iterative application of a linear operator D to a function that is, repeatedly composing D with itself, as in \begin D^n(f) &= (\underbrace_n)(f) \\ &= \underbrace_n (f)\cdots))). \end For example, one may ask for a meaningful interpretation of \sqrt = D^ as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied to any f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differintegral
In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by :\mathbb^q f is the fractional derivative (if ''q'' > 0) or fractional integral (if ''q'' So, \frac = \mathcal^\left\ which generalizes to \mathbb^qf(t) = \mathcal^\left\. Under the bilateral Laplace transform, here denoted by \mathcal and defined as \mathcal (t)=\int_^\infty e^ f(t)\, dt, differentiation transforms into a multiplication \mathcal\left frac\right= s\mathcal (t) Generalizing to arbitrary order and solving for \mathbb^qf(t), one obtains \mathbb^qf(t)=\mathcal^\left\. Representation via Newton series is the Newton interpolation over consecutive integer orders: \mathbb^qf(t) =\sum_^ \binom m \sum_^m\binom mk(-1)^f^(x). For fractional derivative definitions described in this section, the following identities hold: :\mathbb^q(t^n)=\fract^ :\mathbb^q(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \operatorname(z_1), \operatorname(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.. Specifically, see 6.2 Beta Function. A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function that : \Beta(m,n) =\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Formula For Repeated Integration
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress ''n'' antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer ''n'' it yields the definition of fractional integrals and (with ''n'' 0, and the factorial is replaced by the gamma function. The two formulas agree when \alpha \in \Z_. Both the Cauchy formula and the Riemann–Liouville integral are generalized to arbitrary dimensions by the Riesz potential. In fractional calculus, these formulae can be used to construct a differintegral, allowing one to differentiate or integrate a fractional number of times. Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result. References * Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operational Calculus
Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation. History The idea of representing the processes of calculus, differentiation and integration, as operators has a long history that goes back to Gottfried Wilhelm Leibniz. The mathematician Louis François Antoine Arbogast was one of the first to manipulate these symbols independently of the function to which they were applied. This approach was further developed by Francois-Joseph Servois who developed convenient notations. Servois was followed by a school of British and Irish mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester. Treatises describing the application of operator methods to ordinary and partial differential equations w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oliver Heaviside
Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, and rewrote Maxwell's equations in the form commonly used today. He significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death. His formulation of the telegrapher's equations became commercially important during his own lifetime, after their significance went unremarked for a long while, as few others were versed at the time in his novel methodology. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science. Early life Heaviside was born in Camden Town, London, at 55 Kings Street (now Plender Street), the youngest of three children of Thomas, a draughtsman and wood engraver, and Rachel Elizabeth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville
Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse Liouville (née Balland). Liouville gained admission to the École Polytechnique in 1825 and graduated in 1827. Just like Augustin-Louis Cauchy before him, Liouville studied engineering at École des Ponts et Chaussées after graduating from the Polytechnique, but opted instead for a career in mathematics. After some years as an assistant at various institutions including the École Centrale Paris, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France in 1850 and a chair in mechanics at the Faculté des Sciences in 1857. Besides his academic achievements, he was very talented in organisational matters. Liouville founded the ''Journal de Mathématiques Pures et Ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions and the discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" Life Early life Niels Henrik Abel was born prematurely in Nedstrand, Norway, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wallis
John Wallis (; ; ) was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 Wallis served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. He was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics. Biography Educational background * Cambridge, M.A., Oxford, D.D. * Grammar School at Tenterden, Kent, 1625–31. * School of Martin Holbeach at Felsted, Essex, 1631–2. * Cambridge University, Emmanuel College, 1632–40; B.A., 1637; M.A., 1640. * D.D. at Oxford in 1654. Family On 14 March 1645, he married Susanna Glynde ( – 16 March 1687). They had three children: # Anne, Lady Blencowe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1675, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Bernoulli
Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth. Biography Early life Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, and his wife, Margarethe Schongauer, and began studying medicine at University of Basel. His father desired that he study business so that he might take over the family spice trade, but Johann Bernoulli did not like business and convinced his father to allow him to study medicine instead. Johann Bernoulli began studying mathematics on the side with his older brother Jacob Bernoulli. Throughout Johann Bernoulli's education at Basel University, the Bernoulli brothers worked together, spending much of their time studying the newly discovered infinitesimal calculus. They were among t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
