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 126
1.
Absolute Value -- from MathWorld - Weisstein, Eric W.
The absolute value of a real number x is denoted \left\vert{x}\right\vert and defined as the "unsigned" portion of x, \left\vert{x}\right\vert = x\mathop{\rm sgn}\nolimits (x) = \left\{\begin{array}{ll} -x & \mbox{for x\leq 0}\\ x & \mbox{for x\geq 0,} \end{array}\right. where \mathop{\rm sgn}\nolimits (x) is the sign function sgn. The absolute value is therefore always greater than or equal to 0. The absolute value of x for real x is plotted above. The absolute value ...
2.
Absolutely Monotonic Function -- from MathWorld - Weisstein, Eric W.
A function f(x) is absolutely monotonic in the interval a
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Almost Periodic Function -- from MathWorld - Weisstein, Eric W.
A function representable as a generalized Fourier series. Let \mathbb{R} be a metric space with metric \rho(x,y). Following Bohr (1947), a continuous function x(t) for (-\infty0, there exists \ell=\ell(\epsilon)>0 such that every interval [t_0,t_0+\ell(\epsilon)] contains at least one number \tau for which \rho[x(t),x(t+\tau)]<\epsilon for (-\infty
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Arithmetic Function -- from MathWorld - Weisstein, Eric W.
An arithmetic function is a function f(n) defined for all n\in\mathbb{N}, usually taken to be complex-valued, so that f:\mathbb{N}\to\mathbb{C} (Jones and Jones 1998, p. 143). An alternative definition of arithmetic function is a function \psi(n) such that \psi(n+m)=\psi(\psi(n)+\psi(m)) and \psi(n,m)=\psi(\psi(n)\psi(m)). See also: Integer Function, Number Theoretic Function
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Bilinear Function -- from MathWorld - Weisstein, Eric W.
A function of two variables is bilinear if it is linear with respect to each of its variables. The simplest example is f(x,y)=xy. See also: Bilinear Basis, Linear Function, Symmetric Bilinear Form
6.
Bounded -- from MathWorld - Weisstein, Eric W.
A mathematical object (such as a set or function) is said to bounded if it possesses a bound, i.e., a value which all members of the set, functions, etc., are less than. See also: Bounded Operator, Bounded Set
7.
Closed Map -- from MathWorld - Weisstein, Eric W.
A map f between topological spaces that maps closed sets to closed sets. If f is bijective, then f \hbox{ is closed }\Longleftrightarrow f\hbox{ is open } \Longleftrightarrow f^{-1}\hbox{ is continuous,} where f^{-1} denotes the inverse map. In particular, a homeomorphism can be characterized as a continuous bijection which is open (or, equivalently, closed).
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Codomain -- from MathWorld - Weisstein, Eric W.
A set within which the values of a function lie (as opposed to the range, which is the set of values that the function actually takes). See also: Domain, Range
9.
Complete Biorthogonal System -- from MathWorld - Weisstein, Eric W.
A set of functions \{f_1(n,x),f_2(n,x)\} is termed a complete biorthogonal system in the closed interval R if, they are biorthogonal, i.e., \int_R f_1(m,x)f_1(n,x)\,dx = c_m\delta_{mn} \int_R f_2(m,x)f_2(n,x)\,dx = d_m\delta_{mn} \int_R f_1(m,x)f_2(n,x)\,dx = 0 \int_R f_1(m,x)\,dx = 0 \int_R f_2(m,x)\,dx = 0 and complete. A complete biorthogonal system has a very special type of generalized Fourier series. The prototypical example of a complete biorthogonal system is ...
10.
Complete Convex Function -- from MathWorld - Weisstein, Eric W.
A function f(x) is completely convex in an open interval (a, b) if it has derivatives of all orders there and if (-1)^k f^{(2k)}(x) \ge 0 for k = 0, 1, 2, ... in that interval (Widder 1945, p. 177). For example, the functions \sin x and \cos x are completely convex in the intervals (0,\pi) and (-\pi/2,\pi/2) respectively. See also: Completely Monotonic Function
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Complete Orthogonal System -- from MathWorld - Weisstein, Eric W.
A set of orthogonal functions \{\phi_n(x)\} is termed complete in the closed interval x\in[a,b] if, for every piecewise continuous function f(x) in the interval, the minimum square error E_n \equiv \left\Vert{f-(c_1\phi_1+\dots+c_n\phi_n)}\right\Vert^2 (where \left\Vert{f}\right\Vert denotes the L2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite. Symbolically, a set of functions is complete if \lim_{m\to \infty} \int_a^b \left[{f(x) -...
12.
Complete Set of Functions -- from MathWorld - Weisstein, Eric W.
See also: Complete Biorthogonal System, Complete Orthogonal System, Orthogonal Functions, Orthonormal Functions, Overcomplete System
13.
Completely Monotonic Function -- from MathWorld - Weisstein, Eric W.
A completely monotonic function is a function f(x) such that {(-1)}^{-n} f^{(n)}(x) \geq 0 for n = 0, 1, 2, .... Such functions occur in areas such as probability theory (Feller 1971), numerical analysis, and elasticity (Ismail et al. 1986). See also: Complete Convex Function, Monotonic Function
14.
Completely Multiplicative Function -- from MathWorld - Weisstein, Eric W.
A real arithmetic function f(n) is called completely multiplicative (or sometimes totally multiplicative) if f(mn)=f(m)f(n) holds for each pair of integers (m, n). See also: Multiplicative Function
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Complex Map -- from MathWorld - Weisstein, Eric W.
A complex map is a map f:\mathbb{C}\to\mathbb{C}. The following table lists several common types of complex maps. map formula domain inversion f(z) = {1\over z} complex magnification f(z) = aza\in\mathbf{r}\not= 0 magnification+rotation f(z) = aza\in\mathbb{C}\not= 0 Möbius transformation f(z) = {az+b\over cz+d}a, b, c, d \in \mathbb{C} complex rotation f(z) = e^{i\theta}z\theta\in\mathbb{R} complex translation f(z) = z+aa\in\mathbb{C} See also: Complex Rotation, Complex...
16.
Complex Modulus -- from MathWorld - Weisstein, Eric W.
The modulus of a complex number z, also called the complex norm, is denoted \left\vert{z}\right\vert and defined by \left\vert{x+iy}\right\vert \equiv \sqrt{x^2+y^2}. If z is expressed as a complex exponential (i.e., a phasor), then \left\vert{re^{i\phi}}\right\vert = \left\vert{r}\right\vert. The complex modulus is implemented in Mathematica as Abs[z], or as Norm[z] in Mathematica version 5.0. The square \left\vert{z}\right\vert^2 of \left\vert{z}\right\vert is sometimes called the...
17.
Constant Map -- from MathWorld - Weisstein, Eric W.
A map f:X\longrightarrow Y is called constant with constant value y if f(x)=y for all x\in X, i.e., if all elements of X are sent to same element y of Y. See also: Constant Function, Identity Map, Zero Map
18.
Decreasing Function -- from MathWorld - Weisstein, Eric W.
A function f(x) decreases on an interval I if f(b) a, where a,b\in I. Conversely, a function f(x) increases on an interval I if f(b)>f(a) for all b > a with a,b\in I. If the derivative f'(x) of a continuous function f(x) satisfies f'(x)<0 on an open interval (a, b), then f(x) is decreasing on (a, b). However, a function may decrease on an interval without having a derivative defined at all points. For example, the function -x^{1/3} is decreasing everywhere,...
19.
Doubly Periodic Function -- from MathWorld - Weisstein, Eric W.
A function f(z) is said to be doubly periodic if it has two periods \omega_1 and \omega_2 whose ratio \omega_2/\omega_1 is not real. A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function (Whittaker and Watson 1990, p. 429). The periods \omega_1 and \omega_2 play the same part in the theory of elliptic functions as does the single period in the case of the trigonometric...
20.
Elementary Function -- from MathWorld - Weisstein, Eric W.
A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and algebraic, exponential, and logarithmic functions and their inverses under repeated compositions (Shanks 1993, p. 145; Chow 1999). Among the simplest elementary functions are the logarithm, exponential function (including the hyperbolic functions), power function, and trigonometric functions. Following Liouville (1837,...
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