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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, [1] algebra, [2] geometry, [1] and analysis, [3 ...
Look up calculus in Wiktionary, the free dictionary. In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine.
The number π (/paɪ/; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although ...
Glossary of mathematical symbols. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various ...
- 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...
The number e is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of as n tends to infinity, an expression that arises in the computation of compound interest. It is the value at 1 of the (natural) exponential function, commonly ...
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known.
