Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 211
1.
Abel-Plana Formula -- from MathWorld - Weisstein, Eric W.
The Abel-Plana formula gives an expression for the difference between a discrete sum and the corresponding integral. The formula can be derived from the argument principle \oint_\gamma f(z){g'(z)\over g(z)}\,dz = \sum_n f(\mu_n)-\sum_m f(\nu_m), where \mu_n are the zeros of g(z) and \nu_m are the poles contained within the contour \gamma. An appropriate choice of g and \gamma then yields \sum_{n=0}^\infty f(n) - \int_0^\infty f(x)\,dx = {\textstyl... ...}\int_0^\infty [f(it)-f(-it)][\cot(\pi...
2.
Accumulation Point -- from MathWorld - Weisstein, Eric W.
An accumulation point is a point which is the limit of a sequence, also called a limit point. For some maps, periodic orbits give way to chaotic ones beyond a point known as the accumulation point. See also: Bolzano-Weierstrass Theorem, Cantor's Intersection Theorem, Chaos, Fractional Part, Heine-Borel Theorem, Limit Point, Logistic Map, Mode Locking, Period Doubling, Pisot-Vijayaraghavan Constant
5.
Area Integral -- from MathWorld - Weisstein, Eric W.
A double integral over three coordinates giving the area within some region R, A=\mathop{\int\!\!\!\int}\limits _R dx\,dy. If a plane curve is given by y=f(x), then the area between the curve and the x-axis from x = a to x = b is given by A=\int_a^b f(x)\,dx. See also: Integral, Line Integral, Lusin Area Integral, Multiple Integral, Surface Integral, Volume Integral
6.
Asymptotic -- from MathWorld - Weisstein, Eric W.
The term asymptotic means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). A line or curve A that is asymptotic to given curve C is called the asymptote of C. Hardy and Wright (1979, p. 7) use the symbol \asymp to denote that one quantity is asymptotic to another. If f\asymp\phi, then Hardy and Wright say that f and \phi are of the same order of magnitude. See also: Asymptosy, Asymptote, Asymptotic Notation, Asymptotic Curve, Asymptotic Direction,...
7.
Bolzano-Weierstrass Theorem -- from MathWorld - Weisstein, Eric W.
Every bounded infinite set in \mathbb{R}^n has an accumulation point. For n = 1, an infinite subset of a closed bounded set S has an accumulation point in S. For instance, given a bounded sequence a_n, with -C \leq a_n \leq C for all n, it must have a monotonic subsequence a_{n_k}. The subsequence a_{n_k} must converge because it is monotonic and bounded. Because S is closed, it contains the limit of a_{n_k}. The Bolzano-Weierstrass theorem is closely related to the Heine-Borel theorem and...
8.
Bounded Variation -- from MathWorld - Weisstein, Eric W.
A function f(x) is said to have bounded variation if, over the closed interval x\in [a,b], there exists an M such that \left\vert{f(x_1)-f(a)}\right\vert+\left\vert{f(x_2)-f(x_1)}\right\vert+\dots +\left\vert{f(b)-f(x_{n-1})}\right\vert \leq M for all a < x_1 < x_2 < \dots < x_{n-1} < b. The space of functions of bounded variation is denoted "BV," and has the seminorm \Phi(f) = \sup \int f {d\phi\over dx}, where \phi ranges over all compactly supported functions...
9.
Calculus -- from MathWorld - Weisstein, Eric W.
In general, "a" calculus is an abstract theory developed in a purely formal way. "The" calculus, more properly called analysis (or real analysis or, in older literature, infinitesimal analysis) is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects. The calculus is sometimes divided into differential and integral calculus, concerned with derivatives {d\over dx} f(x) ...
10.
Cantor's Intersection Theorem -- from MathWorld - Weisstein, Eric W.
A theorem about (or providing an equivalent definition of) compact sets, originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty closed sets C_1 \supset C_2 \supset C_3 \supset \dots in the real numbers, then Cantor's intersection theorem states that there must exist a point p in their intersection, p\in C_n for all n. For example, 0 \in \cap [0, 1/n]. It is also true in higher dimensions of Euclidean space. Note that the hypotheses stated above are crucial. The...
11.
Catalan Integrals -- from MathWorld - Weisstein, Eric W.
Special cases of general formulas due to Bessel. J_0(\sqrt{z^2-y^2}\,)={1\over\pi}\int_0^\pi e^{y\cos\theta}\cos(z\sin\theta)\,d\theta, where J_0(z) is a Bessel function of the first kind. Now, let z\equiv 1-z' and y\equiv 1+z'. Then J_0(2i\sqrt{z}\,) = {1\over\pi} \int_0^\pi e^{(1+z)\cos\theta}\cos[(1-z)\sin\theta]\,d\theta. See also: Bessel Function of the First Kind
12.
Cauchy Principal Value -- from MathWorld - Weisstein, Eric W.
The Cauchy principal value of a finite integral of a function f about a point c with a\leq c\leq b is given by PV \int_a^b f(x)\,dx \equiv\lim_{\epsilon\to 0^+} \left[{\int^{c-\epsilon}_a f(x)\,dx+\int^b_{c+\epsilon}f(x)\,dx}\right] (Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function f is defined by PV \int^\infty_{-\infty} f(x)\,dx \equiv \lim_{R\to \infty}...
13.
Cauchy's Cosine Integral Formula -- from MathWorld - Weisstein, Eric W.
\int_{-\pi/2}^{\pi/2} \cos^{\mu+\nu-2}\theta e^{i\theta(\mu-... ...over 2^{\mu+\nu-2}\Gamma(\mu+\xi)\Gamma(\nu-\xi)}, where \Gamma(z) is the gamma function.
14.
Cavalieri's Quadrature Formula -- from MathWorld - Weisstein, Eric W.
The definite integral \int_a^b x^n\,dx = \cases{ {b^{n+1}-a^{n+1}\over n+1} & for n\not=1\cr \ln\left({b\over a}\right)& for n=-1,\cr} where a, b, and x are real numbers and \ln x is the natural logarithm. See also: Power
15.
Chain Rule -- from MathWorld - Weisstein, Eric W.
If g(x) is differentiable at the point x and f(x) is differentiable at the point g(x), then f\circ g is differentiable at x. Furthermore, let y=f(g(x)) and u=g(x), then {dy\over dx}={dy\over du} \cdot {du\over dx}. There are a number of related results which also go under the name of "chain rules." For example, if z=f(x,y), x=g(t), and y=h(t), then {dz\over dt}={\partial z\over\partial x}{dx\over dt}+{\partial z\over\partial y}{dy\over dt}. The "general" chain rule...
16.
Change of Variables Theorem -- from MathWorld - Weisstein, Eric W.
A theorem which effectively describes how lengths, areas, volumes, and generalized n-dimensional volumes (contents) are distorted by differentiable functions. In particular, the change of variables theorem reduces the whole problem of figuring out the distortion of the content to understanding the infinitesimal distortion, i.e., the distortion of the derivative (a linear map), which is given by the linear map's determinant. So f:{\mathbb{R}}^n\to{\mathbb{R}}^n is an area-preserving linear...
17.
Closed Interval -- from MathWorld - Weisstein, Eric W.
A closed interval is an interval that includes all of its limit points. If the endpoints of the interval are finite numbers a and b, then the interval \{x:a\leq x\leq b\} is denoted [a, b]. If one of the endpoints is \pm\infty, then the interval still contains all of its limit points, so [a,\infty) and (-\infty, b] are also closed intervals. See also: Closed Ball, Closed Disk, Closed Set, Half-Closed Interval, Interval, Limit Point, Open Interval
18.
Concave Function -- from MathWorld - Weisstein, Eric W.
A function f(x) is said to be concave on an interval [a, b] if, for any points x_1 and x_2 in [a, b], the function -f(x) is convex on that interval (Gradshteyn and Ryzhik 2000). See also: Convex Function
19.
Constant of Integration -- from MathWorld - Weisstein, Eric W.
Since the derivative of a constant is zero, any constant may be added to an indefinite integral (i.e., antiderivative) and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. For this reason, indefinite integrals are often written in the form \int f(x)\,dx = F(x)+C, where C is an arbitrary constant known as the constant of integration. Mathematica returns indefinite integrals without constants of...
20.
Continuity -- from MathWorld - Weisstein, Eric W.
The property of being continuous. See also: Continuity Axioms, Continuity Correction, Continuity Principle, Continuous Distribution, Continuous Function, Continuous Space, Fundamental Continuity Theorem, Limit
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