Second derivativeIn calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.
Partial fraction decompositionIn algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms.
Analytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x 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. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Real-valued functionIn mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Let be the set of all functions from a set X to real numbers .
One-form (differential geometry)In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space. Symbolically, where is linear. Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates: where the are smooth functions.
Function of several complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi.
Generalized continued fractionIn complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.
Gauss's continued fractionIn complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions. Lambert published several examples of continued fractions in this form in 1768, and both Euler and Lagrange investigated similar constructions, but it was Carl Friedrich Gauss who utilized the algebra described in the next section to deduce the general form of this continued fraction, in 1813.
Continued fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression.
Mathematical economicsMathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Periodic continued fractionIn mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,...ak+m] of partial denominators that repeats ad infinitum. For example, can be expanded to a periodic continued fraction, namely as [1,2,2,2,...]. The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem.
