搜尋結果
Hang Seng China 50 Index ( Chinese: 恒生神州50指數) is a pan-China stock market index to represent the top 50 China-based companies in the stock exchanges of Hong Kong, Shanghai and Shenzhen, which covers A share (shares circulated in mainland China), H share (shares circulated in Hong Kong from the mainland China incorporated ...
The PHLX Semiconductor Sector ( SOX) is a Philadelphia Stock Exchange capitalization-weighted index composed of the 30 largest U.S. companies primarily involved in the design, distribution, manufacture, and sale of semiconductors. It was created in 1993 by the Philadelphia Stock Exchange. The Index contains the following 30 companies as updated ...
- Graph
- Relation to More General Exponential Functions
- Formal Definition
- Overview
- Derivatives and Differential Equations
- Continued Fractions For Ex
- Complex Plane
- Matrices and Banach Algebras
- Lie Algebras
- Transcendency
The graph of y = e x {\displaystyle y=e^{x}} is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangen...
The exponential function f ( x ) = e x {\displaystyle f(x)=e^{x}} is sometimes called the natural exponential function in order to distinguish it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive b, As functions of a real ...
The real exponential function exp : R → R {\displaystyle \exp :\mathbb {R} \to \mathbb {R} } can be characterized in a variety of equivalent ways. It is commonly defined by the following power series: Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers; see § Complex plane...
The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683to the number If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the inter...
The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is, Functions of the form cex for constant c are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of sa...
A continued fraction for ex can be obtained via an identity of Euler: The following generalized continued fraction for ezconverges more quickly: or, by applying the substitution z = x/y: This formula also converges, though more slowly, for z > 2. For example:
As in the real case, the exponential function can be defined on the complex planein several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: Alternatively, the complex exponential function may be defined by...
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommutin...
Given a Lie group G and its associated Lie algebra g {\displaystyle {\mathfrak {g}}} , the exponential map is a map g {\displaystyle {\mathfrak {g}}} ↦ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a spe...
The function ez is not in the rational function ring C ( z ) {\displaystyle \mathbb {C} (z)} : it is not the quotient of two polynomials with complex coefficients. If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over C ( z ) {\displaystyle \mathbb {C} (z)} , and hence ez is transcendental over C ( z ) {\di...
Índice Bovespa. The Bovespa Index ( Portuguese: Índice Bovespa ), best known as Ibovespa is the benchmark index of about 86 stocks [1] traded on the B3 (Brasil Bolsa Balcão), accounting for the majority of trading and market capitalization in the Brazilian stock market. It is a weighted measurement index.
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has. where e is the base of the natural logarithm, i is the imaginary unit, and ...
e. In economics, the Gini coefficient ( / ˈdʒiːni / JEE-nee ), also known as the Gini index or Gini ratio, is a measure of statistical dispersion intended to represent the income inequality, the wealth inequality, or the consumption inequality [3] within a nation or a social group. It was developed by Italian statistician and sociologist ...
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary ), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including ...
