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Frullani Integral
In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani. The integrals are of the form :\int _^\,x where f is a function defined for all non-negative real numbers that has a limit at \infty, which we denote by f(\infty). The following formula for their general solution holds if f is continuous on (0,\infty), has finite limit at \infty, and a,b > 0: :\int _^\,x=\Big(f(\infty)-f(0)\Big)\ln . If f (\infty) does not exist, but \int_c^\infty \frac dx exists for some c > 0, then \int _^\,x= -f(0)\ln . Proof for continuously differentiable functions A simple proof of the formula (under stronger assumptions than those stated above, namely f \in \mathcal^1(0,\infty)) can be arrived at by using the Fundamental theorem of calculus to express the integrand In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the pr ... [...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] |
Improper Integral
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge. If a regular definite integral (which may retronymically be called a proper integral) is worked out as if it is improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darbou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuliano Frullani
People with the Italian given name or surname Giuliano () have included: Origin and meaning Giuliano is an Italian form of Julian, a masculine name meaning "youthful". A patronymic Italian surname derived from Giuliano is Giuliani. In arts and entertainment Surname * Gaetano Giuliano (1929–2023), Italian politician * Geoffrey Giuliano (born 1953), American author * Maurizio Giuliano (born 1975), Italian-British writer and Guinness-record-holding traveller Given name * Giuliano Pesello (ca. 1367–1446), Italian painter * Giuliano Gemma (1938–2013), Italian actor * Giuliano Đanić (born 1973), Croatian pop-folk singer In crime * Luigi Giuliano (born 1949), former Neapolitan Camorra boss and pentito * Salvatore Giuliano (1922–1950), Sicilian bandit In politics Surname * Carla Giuliano (born 1983), Italian MP * Neil Giuliano (born 1956), former mayor of Tempe, Arizona, US * Sebastian Giuliano (born 1952), mayor of Middletown, Connecticut, US Given name * Giuliano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Function
Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, the output value can be made ''arbitrarily'' close to if the input to is taken ''sufficiently'' close to . On the other hand, if some inputs very close to are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''. The notion of a limit has many applications in modern calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be thought of as inverses of each other. The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function , an antiderivative or indefinite integral can be obtained as the integral of over an interval with a variable upper bound. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed Interval (mathematics), interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a definite integral pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrand
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''antiderivative'', a function whose derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini's Theorem
In mathematical analysis, Fubini's theorem characterizes the conditions under which it is possible to compute a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable on a rectangle X\times Y, then one can evaluate the double integral as an iterated integral:\, \iint\limits_ f(x,y)\,\text(x,y) = \int_X\left(\int_Y f(x,y)\,\texty\right)\textx=\int_Y\left(\int_X f(x,y) \, \textx \right) \texty. This formula is generally not true for the Riemann integral, but it is true if the function is continuous on the rectangle. In multivariable calculus, this weaker result is sometimes also called Fubini's theorem, although it was already known by Leonhard Euler. Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli theorems are usually combined and for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Master Theorem
In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function f(x) has an expansion of the form f(x)=\sum_^\infty \frac(-x)^k then the Mellin transform of f(x) is given by \int_0^\infty x^ f(x) \, dx = \Gamma(s)\,\varphi(-s) where \Gamma(s) is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams. A similar result was also obtained by Glaisher. Alternative formalism An alternative formulation of Ramanujan's master theorem is as follows: \int_0^\infty x^\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \frac\,\lambda(-s) which gets converted to the above form after substituting \lambda(n) \equiv \frac and us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
