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Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Growth of quantities at rate proportional to the current amount
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size
Exponential_growth
Mathematical function common in physics
The stretched exponential function f β ( t ) = e − t β {\displaystyle f_{\beta }(t)=e^{-t^{\beta }}} is obtained by inserting a fractional power law into
Stretched exponential function
Stretched_exponential_function
Exponential function of an exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f ( x ) = a b x = a ( b x ) {\displaystyle
Double_exponential_function
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Special function defined by an integral
In mathematics, the exponential integral E i {\displaystyle \mathrm {Ei} } is a special function on the complex plane. It is defined as one particular
Exponential_integral
Functional square root of an exponential
In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f {\displaystyle f} such that f
Half-exponential_function
Probability distribution
gamma, and Poisson distributions. The probability density function (pdf) of an exponential distribution is f ( x ; λ ) = { λ e − λ x x ≥ 0 , 0 x < 0.
Exponential_distribution
Arithmetic operation
tetration in Wiktionary, the free dictionary. Ackermann function Big O notation Double exponential function Hyperoperation Iterated logarithm Symmetric level-index
Tetration
2.71828…, base of natural logarithms
equal to 2.71828, that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician
E_(mathematical_constant)
exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integrals are antiderivative functions
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
Mathematical concept
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes
Characterizations of the exponential function
Characterizations_of_the_exponential_function
Mathematical function, inverse of an exponential function
to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete
Logarithm
Arithmetic operation
integer Mathematics portal Double exponential function – Exponential function of an exponential function Exponential decay – Decrease in value at a rate
Exponentiation
Generates a forecast of future values of a time series
window function. Whereas in the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing
Exponential_smoothing
Hyperbolic analogues of trigonometric functions
With hyperbolic angle u, the hyperbolic functions sinh and cosh can be defined with the exponential function eu. In the figure A = ( e − u , e u ) ,
Hyperbolic_functions
Topics referred to by the same term
to exponentiation, including: Exponential function, also: Matrix exponential, the matrix analogue to the above Exponential decay, decrease at a rate proportional
Exponential
Mathematical function
particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex
P-adic_exponential_function
Mathematical field with an extra operation
an exponential field, and the function E {\displaystyle E} is called an exponential function on F {\displaystyle F} . Thus an exponential function on
Exponential_field
Logarithm to the base of the mathematical constant e
real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities: e ln x = x if x ∈ R + ln
Natural_logarithm
Family of probability distributions related to the normal distribution
single-parameter exponential family is a set of probability distributions whose probability density function (or probability mass function, for the case
Exponential_family
Mathematical approximation of a function
all x. The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function: exp ( exp
Taylor_series
Fundamental trigonometric functions
definition of both sine and cosine functions can be extended in a complex plane in terms of an exponential function as follows: sin ( θ ) = e i θ − e
Sine_and_cosine
Functions of an angle
sin and cos can be defined for all complex numbers in terms of the exponential function, via power series, or as solutions to differential equations given
Trigonometric_functions
model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred
Tarski's exponential function problem
Tarski's_exponential_function_problem
Map from a Lie algebra to its Lie group
the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical
Exponential_map_(Lie_theory)
Formal power series
are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and
Generating_function
Probability of survival beyond any specified time
failures is approximated using the exponential function, then the exponential curve gives the probability density function, fT, for AC failure times. Another
Survival_function
Association of one output to each input
algebraic function is the same, with nth roots and roots of polynomials also allowed. An elementary function is the same, with logarithms and exponential functions
Function_(mathematics)
Complex exponential in terms of sine and cosine
fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x
Euler's_formula
Branch of mathematics studying functions of a complex variable
complex functions are defined in this way, including the complex exponential function, complex logarithm functions, and trigonometric functions. Complex
Complex_analysis
Rate-seeking algorithm
rate (i.e., back off). The rate reduction can be modelled as an exponential function: t = b c {\displaystyle t=b^{c}} or f = 1 b c {\displaystyle f={\frac
Exponential_backoff
Function that is holomorphic on the whole complex plane
functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine
Entire_function
types of functions Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) Algebraic functions are functions
List of mathematical functions
List_of_mathematical_functions
Analytic function that does not satisfy a polynomial equation
contrast to an algebraic function. The most familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses
Transcendental_function
Exponentation in modular arithmetic
exponent e when given b, c, and m – is believed to be difficult. This one-way function behavior makes modular exponentiation a candidate for use in cryptographic
Modular_exponentiation
Infinite sum of monomials
one of the most important examples of a power series, as are the exponential function formula e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ {\displaystyle
Power_series
Real function with secant line between points above the graph itself
number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and an exponential function c e x {\displaystyle
Convex_function
Number with a real and an imaginary part
be regarded as its norm.] However for another inverse function of the complex exponential function (and not the above defined principal value), the branch
Complex_number
Type of mathematical function
functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse
Elementary_function
Function used in signal processing
exponential function never reaches zero, the values of the window at its limits are non-zero (it can be seen as the multiplication of an exponential function by
Window_function
Product of numbers from 1 to n
analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory
Factorial
Generalization of the real numbers
real exponential function exp(x) (with base e) to the surreals was carried through by Gonshor. The powers of ω function is also an exponential function, but
Surreal_number
Multivalued function in mathematics
is any complex number and e w {\displaystyle e^{w}} is the exponential function. The function is named after Johann Lambert, who considered a related problem
Lambert_W_function
S-shaped curve
{\displaystyle L} . The exponential function with negated argument ( e − x {\displaystyle e^{-x}} ) is used to define the standard logistic function where L = 1
Logistic_function
Logarithm of a complex number
to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense. There
Complex_logarithm
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
Topics referred to by the same term
the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include: exponential map
Exponential_map
Finite sum formed using the exponential function
an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually
Exponential_sum
Type of complex function with growth bounded by an exponential function
mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function e C | z | {\displaystyle e^{C|z|}}
Exponential_type
Example of the learning curve effect on performance
individual-level data is better fit by an exponential function and the authors demonstrate that the multiple exponential curves will average to produce a curve
Power_law_of_practice
Swiss mathematician (1707–1783)
harmonic series, the gamma function, and values of the Riemann zeta function. Euler introduced the use of the exponential function and logarithms in analytic
Leonhard_Euler
Constant e raised to the power of pi
the exponential of pi eπ, also called Gelfond's constant, is the real number e raised to the power π (i.e., the value of the exponential function at π)
Gelfond's_constant
the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm. Trigonometric functions may be deduced from
List of trigonometric identities
List_of_trigonometric_identities
Type of activation function
the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models
Rectified_linear_unit
exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy
List_of_exponential_topics
Function with a repeating pattern
Functions with a domain in the complex numbers can exhibit more complex periodic properties. The complex exponential function is a periodic function with
Periodic_function
Solution method for linear differential equations
calculation in quantum mechanics in which the wave function is recast as an exponential function, semiclassically expanded, and then either the amplitude
WKB_approximation
Special function in the physical sciences
{\displaystyle k<0} and exponential for k > 0 {\displaystyle k>0} , the Airy functions are oscillatory for x < 0 {\displaystyle x<0} and exponential for x > 0 {\displaystyle
Airy_function
Mathematical formula involving a given set of operations
Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set
Closed-form_expression
Array in complex analysis
function f(z) exists, then the main diagonal of the Padé table representing f(z) is normal. Here is an example of a Padé table, for the exponential function
Padé_table
The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is
Exponential_response_formula
Extension of the factorial function
exponential decay: Γ ( z ) = M { e − x } ( z ) . {\displaystyle \Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.} Other extensions of the factorial function
Gamma_function
N-th root of the product of n numbers
logarithms, and then returning the result to linear scale by using the exponential function exp {\displaystyle \exp } , a 1 a 2 ⋯ a n t n = exp ( ln a
Geometric_mean
Mathematical function
chemistry to form basis sets. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ( α x 2 + β x
Gaussian_function
Type of function in mathematics
analytic function whose zeros have an accumulation point must vanish identically. Standard examples include polynomials, the exponential function, and the
Analytic_function
exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function
Exponential_polynomial
Mathematical equation linking e, i and π
defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common
Euler's_identity
Statistical physics approach
generated by κ-deformed functions, especially the κ-exponential function. The Kaniadakis exponential (or κ-exponential) function is a one-parameter generalization
Kaniadakis_statistics
Q-analog in combinatorial mathematics
in elsewhere. In combinatorial mathematics, a q-exponential is a q-analog of the exponential function, namely the eigenfunction of a q-derivative. There
Q-exponential
mathematics, the Carlitz exponential (named after Leonard Carlitz) is a characteristic p analogue to the usual exponential function studied in real and complex
Carlitz_exponential
Advanced Placement course and exam
and science courses. In this course, students study a broad spectrum of function types that are foundational for careers in mathematics, physics, biology
AP_Precalculus
Hypothetical event
alleged to mistake the logistic function (S-function) for an exponential function, and to see a "knee" in an exponential function where there can in fact be
Technological_singularity
Mathematical conjecture
multiplication, and some special meromorphic transcendental functions (e.g. exponential or modular functions) have solutions in the complex numbers. This question
Existential closedness conjecture
Existential_closedness_conjecture
is binomial coefficient exp ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle x} See Faulhaber's formula. ∑ k = 0 m k n − 1 = B
List_of_mathematical_series
transition function include exponential function and first and second-order logistic functions. They give rise to Logistic STAR (LSTAR) and Exponential STAR
STAR_model
Type of computer arithmetic
being invertible on this interval. The inverse, the generalized exponential function, is defined by φ ( x ) = { x if 0 ≤ x < 1 , e φ ( x − 1 ) if x
Symmetric level-index arithmetic
Symmetric_level-index_arithmetic
Quantum search algorithm
of an exponential function is still an exponential, not a polynomial function). Unlike other quantum algorithms, which may provide exponential speedup
Grover's_algorithm
for simple theories Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable? The universality
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Formula for estimating hiking speed
Tobler's hiking function is an exponential function determining the hiking speed, taking into account the slope angle. It was formulated by Waldo Tobler
Tobler's_hiking_function
Strong form of uniform continuity
the first property below. Analytic functions that are not (globally) Lipschitz continuous The exponential function becomes arbitrarily steep as x → ∞
Lipschitz_continuity
Mathematical function such that every output has at least one input
positive real numbers to the set of all real numbers). Its inverse, the exponential function, if defined with the set of real numbers as the domain and the codomain
Surjective_function
Generalization of the exponential function
one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient
C0-semigroup
Collection of mathematical functions originated by Constantino Tsallis
proposed by George Box and David Cox in 1964. The q-exponential is a deformation of the exponential function using the real parameter q. e q ( x ) = { exp
Tsallis_statistics
Shift-and-add algorithm
Jean-Michel Muller. BKM is based on computing complex logarithms (L-mode) and exponentials (E-mode) using a method similar to the algorithm Henry Briggs used to
BKM_algorithm
Approximation of a function by a polynomial
transcendental functions such as the exponential function and trigonometric functions. It is the starting point of the study of analytic functions, and is fundamental
Taylor's_theorem
Types of special mathematical functions
)}}\end{aligned}}} is the limiting function to the upper incomplete gamma function as s → 0, also known as the exponential integral E 1 ( z ) {\displaystyle
Incomplete_gamma_function
Differential equation that is linear with respect to the unknown function
and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric
Linear_differential_equation
Statistical model for a binary dependent variable
equivalent to the exponential function of the linear regression expression. This illustrates how the logit serves as a link function between the probability
Logistic_regression
Number of partitions of an integer
grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's
Partition function (number theory)
Partition_function_(number_theory)
Probability distribution
distributions not only forms an exponential family (EF), but in fact forms a natural exponential family (NEF) with quadratic variance function (NEF-QVF). Many properties
Normal_distribution
elementary recursive function. Equivalently, these are the problems that can be solved in time bounded by an iterated exponential function with a bounded number
ELEMENTARY
bump A bump function is a nonzero compactly-supported smooth function, usually constructed using the exponential function. BV A BV-function or a bounded
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions List
Lists_of_integrals
Time for exponential decay to remove half of a quantity
is also used more generally to characterize any type of exponential (or, rarely, non-exponential) decay. For example, the medical sciences refer to the
Half-life
mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition
Plethystic_exponential
Function increasing at a decreasing rate of increase
mathematical function is constantly increasing at a decreasing rate. Asymptotically, bounded growth approaches a fixed value. This contrasts with exponential growth
Bounded_growth
Mathematical functions having established names and notations
cosine ( cos {\displaystyle \cos } ), exponential function ( exp {\displaystyle \exp } ), and error function ( erf {\displaystyle \operatorname {erf}
Special_functions
Polynomial function of degree at most one
of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x))
Linear_function_(calculus)
Sum of elements on the main diagonal
eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant: det ( exp ( A ) ) = exp ( tr
Trace_(linear_algebra)
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, a high Egyptian functionary.
Biblical
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Male
Egyptian
, the son of the functionary Heknofre.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
Surname or Lastname
English
English : occupational name for a house servant who attended his master in his private quarters (see Bower 2).Americanized spelling of German Bauermann, a variant of Bauer.
Boy/Male
American, Australian, British, English
From the Enclosed Town
Girl/Female
American, Australian, British, English, French
Sorrowful; From Lorraine
Boy/Male
Hebrew American Biblical
God has helped.
Girl/Female
Hindu
Full of desires
Boy/Male
British, Chinese, Christian, English, Norse, Scandinavian, Scottish
A Marshland; From the Swampy Place; Man of Strength
Boy/Male
Tamil
To regin universally
Boy/Male
Australian, Polish
Fortunate; Blessed
Boy/Male
Sikh
The Sun
Boy/Male
Hawaiian Spanish
Shark. A passionate lover.
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
EXPONENTIAL FUNCTION
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
a.
Pertaining to, or connected with, a function or duty; official.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
a.
Of, pertaining to, or designating, certain secret tribunals which flourished in Germany from the end of the 12th century to the middle of the 16th, usurping many of the functions of the government which were too weak to maintain law and order, and inspiring dread in all who came within their jurisdiction.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
v. t.
To assign to some function or office.
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
pl.
of Functionary
adv.
In a functional manner; as regards normal or appropriate activity.
a.
Destitute of function, or of an appropriate organ. Darwin.
v. i.
Alt. of Functionate
a.
Pertaining to the function of an organ or part, or to the functions in general.
a.
Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.