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Gautschi's Inequality
In real analysis, a branch of mathematics, Gautschi's inequality is an inequality for ratios of gamma functions. It is named after Walter Gautschi. Statement Let x be a positive real number, and let s\in (0,1). Then, :x^ < \frac < (x + 1)^. History In 1948, Wendel proved the inequalities : for and . He used this to determine the asymptotic behavior of a ratio of gamma functions. The upper bound in this inequality is stronger than the one given above. In 1959, Gautschi independently proved two inequalities for ratios of gamma functions. His lower bounds were identical to Wendel's. One of his upper bounds was the one given in the statement above, while the other one was sometimes stronger and sometimes weaker than Wendel's.Consequences An immediate consequence is the following description of the asymptotic behavior of ratios of gamma functions ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the (established) real number system. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and \cdot, and a total order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique '' complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completenes ... [...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] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than sign, less-than and greater-than sign, greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or no ... [...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] |
Walter Gautschi
Walter Gautschi (; ; born December 11, 1927) is a Swiss-born American mathematician, writer and professor emeritus of Computer science and Mathematics at Purdue University in West Lafayette, Indiana. He is primarily known for his contributions to numerical analysis and has authored over 200 papers in his area and published four books. Early life and education Gautschi was born December 11, 1927, in Basel, Switzerland, to Heinrich Gautschi (1901-1975). His paternal family originally hailed from Reinach. His patrilineal uncle, Adolf Eduard Gautschi, was a custodian and landscape painter. He had one twin brother Werner (1927-1959). He completed a Ph.D. in mathematics from the University of Basel on the thesis ''Analyse graphischer Integrationsmethoden'' advised by Alexander Ostrowski and Andreas Speiser (1953). Career Since then, he did postdoctoral work as a Janggen-Pöhn Research, Fellow at the ''Istituto Nazionale per le Applicazioni del Calcolo'' in Rome (1954) and at the H ... [...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] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Real Numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used for either of these, the notation \R_ or \R^ for \left\ and \R_^ or \R^_ for \left\ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, \R_ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = , z, \mathrm^, with argument \varphi = 0. Properties The set \R_ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Value Theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval (mathematics), interval starting from local hypotheses about derivatives at points of the interval. History A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
