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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
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
Topics referred to by the same term
Double exponential may refer to: A double exponential function Double exponential time, a task with time complexity roughly proportional to such a function
Double_exponential
Arithmetic operation
tetration in Wiktionary, the free dictionary. Ackermann function Big O notation Double exponential function Hyperoperation Iterated logarithm Symmetric level-index
Tetration
Topics referred to by the same term
the cumulative distribution function of which is an iterated exponential function (the exponential of an exponential function). This disambiguation page
Double exponential distribution
Double_exponential_distribution
Generates a forecast of future values of a time series
Exponential smoothing or exponential moving average (EMA) is a technique for smoothing time series data using the exponential window function. Whereas
Exponential_smoothing
Mathematical function, inverse of an exponential function
W function, and the logit. They are the inverse functions of the double exponential function, tetration, of f(w) = wew, and of the logistic function, respectively
Logarithm
the exponential function Catenary Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double exponential
List_of_exponential_topics
Arithmetic operation
an integer Mathematics portal Double exponential function – Exponential function of an exponential function Exponential decay – Decrease in value at a
Exponentiation
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
Number of form 2^(2^p-1)-1 with prime exponent
this number is known as a "Martian prime". Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime Chris Caldwell,
Double_Mersenne_number
Product of numbers from 1 to n
As a function of n {\displaystyle n} , the factorial has faster than exponential growth, but grows more slowly than a double exponential function. Its
Factorial
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
Formal power series
are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and
Generating_function
Decrease in value at a rate proportional to the current value
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by
Exponential_decay
Everything in space and time
nearest so-called doppelgänger is 1010115 metres away from us (a double exponential function larger than a googolplex). The physical basis of these ideas
Universe
Numerical integration method
transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size
Tanh-sinh_quadrature
Function with two complex number "periods"
functions with a single period on the real number line include the trigonometric functions like cosine and sine, In the complex plane the exponential
Doubly_periodic_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
Estimate of time taken for running an algorithm
T(n)} is upper bounded by the factorial function n ! {\displaystyle n!} . Factorial time is a subset of exponential time (EXP) because n ! ≤ n n = 2 n log
Time_complexity
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
Class of statistical models
convenient if V follows from an exponential family of distributions, but it may simply be that the variance is a function of the predicted value. The unknown
Generalized_linear_model
Topics referred to by the same term
in probability. Tetration, also known as hyperexponentiation. Double exponential function This disambiguation page lists articles associated with the title
Hyperexponential
Positive integer of the form (2^(2^n))+1
polygons are constructible partially depends on Fermat primes. Double exponential function Lucas' theorem Mersenne prime Pierpont prime Primality test Proth's
Fermat_number
infra-exponential or subexponential if it is dominated by all exponential growth rates, however great the doubling time. A continuous function with infra-exponential
Infra-exponential
Probability distribution
Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location
Laplace_distribution
Complexity class
module satisfying an LTL specification) is 2EXPTIME-complete. Double exponential function Christos Papadimitriou, Computational Complexity (1994), ISBN 978-0-201-53082-7
2-EXPTIME
Sigmoid shape special function
erf(x) is an odd function, so erf(x) = −erf(−x). Exponential bounds and a pure exponential approximation for the complementary error function are given by
Error_function
Describes the sum of independent normal and exponential random variables
shifted exponential with the weight being a function of the normal distribution. The probability density function (pdf) of the exponentially modified
Exponentially modified Gaussian distribution
Exponentially_modified_Gaussian_distribution
Map from tangent space to the manifold
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M {\displaystyle T_{p}M} of a Riemannian manifold (or pseudo-Riemannian
Exponential map (Riemannian geometry)
Exponential_map_(Riemannian_geometry)
Time for exponential decay to remove half of a quantity
other chemicals in the human body. The converse of half-life is doubling time, an exponential property which increases by a factor of 2 rather than reducing
Half-life
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
Swedish oceanographer (1909–1990)
(2010). Fitting measured irradiance of Jerlov water types to double exponential functions using R. Luxembourg: Publications Office. ISBN 978-92-79-18924-1
Nils_Jerlov
Finite sum of distinct unit fractions
representations of the number one is bounded above and below by double exponential functions of n {\displaystyle n} . The number of different Egyptian fraction
Egyptian_fraction
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
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
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and logarithms with arbitrary
CORDIC
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
Integral of the Gaussian function, equal to sqrt(π)
of Gaussian functions Common integrals in quantum field theory Normal distribution List of integrals of exponential functions Error function Berezin integral
Gaussian_integral
Relationship between solar irradiance and photosynthesis
under excessive irradiance, Steele proposed using a linear function is multiplied by an exponential decay term—a gamma distribution functional type with shape
PI_curve
Mathematical function relating circular and hyperbolic functions
{gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s
Gudermannian_function
Unproven computational hardness assumption
algorithms for larger k {\displaystyle k} take exponential time, with the base of the exponential function depending on k {\displaystyle k} . For instance
Exponential_time_hypothesis
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
single-sided function. Use an exponential function as the impulse response for the support region of positive values as before. In this double-sided filter
FIR_transfer_function
Generalization of the Euler gamma function and the Barnes G-function
gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was
Multiple_gamma_function
Generalized function whose value is zero everywhere except at zero
independent entity. A rigorous interpretation of the exponential form and the various limitations upon the function f {\displaystyle f} necessary for its application
Dirac_delta_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
2005 non-fiction book by Ray Kurzweil
makes the industry grow even faster, effectively an example of "double" exponential growth. Kurzweil dictates evolutionary progress looks smooth, but
The_Singularity_Is_Near
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
Arbitrary-precision calculator supporting interactive and scripted use
contains functions for calculating sine, cosine, arctangent, natural logarithm, the exponential function and the two parameter Bessel function J. Most
Bc_(programming_language)
Formula in Lie group theory
Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to
Derivative of the exponential map
Derivative_of_the_exponential_map
Compounding sum paid for the use of money
up interest in Wiktionary, the free dictionary. Credit card interest Exponential growth Fisher equation Interest rate Rate of return Rate of return on
Compound_interest
Function computable with bounded loops
and exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is
Primitive_recursive_function
Operation on formal power series
ordinary generating function (OGF) of the sequence, denoted F ( z ) {\displaystyle F(z)} , and the exponential generating function (EGF) of the sequence
Generating function transformation
Generating_function_transformation
Concept in machine learning
Double descent in statistics and machine learning is the phenomenon where a model's error rate on the test set initially decreases with the number of parameters
Double_descent
Methods of estimating the doubling time of an investment
periods required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are
Rule_of_72
Topic in probability theory and statistics
(1,ν) random variable. If X is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ. A geometric
Relationships among probability distributions
Relationships_among_probability_distributions
C function to format and output text
standard library function and is also a Linux terminal (shell) command that formats text and writes it to standard output. The function accepts a format
Printf
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
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
Superimposition of two or more exposures to create a single image
to use an exponential decay in which the current frame has the strongest weight, and previous frames are faded out with a sliding exponential window. Multiple
Multiple_exposure
Mathematical functions
complex exponential function. An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see
Lemniscate_elliptic_functions
trigonometric functions, inverse trigonometric functions, logarithm functions, exponential function, hyperbolic functions, Inverse hyperbolic functions, Bell
MXparser
Operation in calculus
function does not have integrals that can be expressed in closed form involving only elementary functions, include rational and exponential functions
Integral
Summatory function of the divisor-counting function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic
Divisor_summatory_function
Function that maps matrices to matrices
entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems
Analytic_function_of_a_matrix
C standard library header file
functions that work on float and long double arguments. Those functions are identified by f and l suffixes respectively. C99 adds several functions and
C_mathematical_functions
Quickly growing function
recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi- and superfactorial functions, and
Ackermann_function
Family of continuous probability distributions
simulation studies since its probability density function, cumulative distribution function and quantile functions can be expressed in closed form. This distribution
Kumaraswamy_distribution
Empirical law on the variance of species in a habitat
of exponential dispersion models. Chapman & Hall. London Rayner, JMV (1985). "Linear relations in biomechanics: the statistics of scaling functions". Journal
Taylor's_law
Hash function that is suitable for use in cryptography
secure cryptographic hash functions but do not usually have a strong connection to practical security. For example, an exponential-time algorithm can sometimes
Cryptographic_hash_function
Traditional Chinese game of four players
function being a piecewise function: a constant amount is given for scoreless hands, and the score is doubled for each point (that is, an exponential
Scoring_in_Mahjong
Conditions for switching order of integration in calculus
states that if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y} , then one can evaluate the double integral as an iterated
Fubini's_theorem
Economic theory
theories such as the concept of exponential growth. It is commonly understood that growth will not continue to rise exponentially, rather it is subject to different
Diminishing_returns
Mathematical equation related to human death rate
components: an age-dependent term, the Gompertz function, that increases approximately exponentially with age, and an approximately age-independent background
Gompertz–Makeham law of mortality
Gompertz–Makeham_law_of_mortality
Particular case of the generalized extreme value distribution
distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer
Gumbel_distribution
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 with no sudden changes
show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous. A right-continuous function A left-continuous
Continuous_function
First pocket scientific calculator
scientific pocket calculator: a calculator with trigonometric and exponential functions. It was introduced in 1972. In about 1970 HP co-founder Bill Hewlett
HP-35
Probability distribution
cumulative distribution function of an exponential distribution with rate α. Pareto distribution can be constructed by hierarchical exponential distributions.
Pareto_distribution
Observation that the intrinsic rate of human performance is exponential
law is the observation that the intrinsic rate of human performance is exponential.[further explanation needed] The law is named after Douglas Engelbart
Engelbart's_law
Stochastic control System dynamics, system analysis Takens' theorem Exponential dichotomy Liénard's theorem Krylov–Bogolyubov theorem Krylov-Bogoliubov
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Mathematical function
In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers
Dickman_function
Functional relationship between two quantities
{\displaystyle i-1,i} . A power law with an exponential cutoff is simply a power law multiplied by an exponential function: f ( x ) ∝ x − α e − β x . {\displaystyle
Power_law
(bioinformatics) Dot plot (statistics) Double counting (fallacy) Double descent Double exponential distribution (disambiguation) Double mass analysis Doubly stochastic
List_of_statistics_articles
Group theory function
no. 2, pp. 467–518. S. M. Gersten, The double exponential theorem for isodiametric and isoperimetric functions. International Journal of Algebra and Computation
Dehn_function
Conjecture in graph theory
whenever the two functions define a homomorphism from G × K2 (the bipartite double cover of G) to K. The exponential graph is the exponential object in the
Hedetniemi's_conjecture
Special functions of several complex variables
We see that the theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore
Theta_function
Quantum mechanical phenomenon
corrections have little quantitative effect. The wave function is expressed as the exponential of a function: Ψ ( x ) = e Φ ( x ) , {\displaystyle \Psi (x)=e^{\Phi
Quantum_tunnelling
Growth of bacterial colonies
conditions, cyanobacteria can double their population four times a day and then they can triple their population. Exponential growth cannot continue indefinitely
Bacterial_growth
Complex-differentiable (mathematical) function
\mathbb {C} } ), and so are the exponential function exp z {\displaystyle \exp z} and the trigonometric functions cos z = 1 2 ( exp ( + i z
Holomorphic_function
Describes approximate behavior of a function
c} is called superpolynomial. One that grows more slowly than any exponential function of the form c n {\displaystyle c^{n}} with c > 1 {\displaystyle c>1}
Big_O_notation
Mathematical function
uppercase form of the archaic Greek letter digamma meaning double-gamma). The gamma function obeys the equation Γ ( z + 1 ) = z Γ ( z ) . {\displaystyle
Digamma_function
Type of matrix
the rotation matrix and the axis-angle format are related by the exponential function. One can derive a simple expression for the generator G. One starts
Infinitesimal_rotation_matrix
Event that is both extremely large in effect and of unique origins
power law tail is exponential, one can take the logarithm of power law data and then test for outliers relative to an exponential tail. There are many
Dragon_king_theory
Gradual change in level of audio signal
exaggerated version of an exponential fade in terms of the apparent volume. Thus, the impression that would be gathered from an exponential curve's fade would
Fade_(audio_engineering)
numbers. Transcendental functions such as log, exponential, and trig functions make up the backbone of any math library. These functions are generally implemented
Math_library
Count of the possible partitions of a set
can be derived by expanding the exponential generating function using the Taylor series for the exponential function, and then collecting terms with the
Bell_number
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