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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
Half-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
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
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
Time for exponential decay to remove half of a quantity
characterize any type of exponential (or, rarely, non-exponential) decay. For example, the medical sciences refer to the biological half-life of drugs and other
Half-life
Function that, applied twice, gives another function
↦ f(x)². The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950, later
Functional_square_root
exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy
List_of_exponential_topics
Arithmetic operation
tetration in Wiktionary, the free dictionary. Ackermann function Big O notation Double exponential function Hyperoperation Iterated logarithm Symmetric level-index
Tetration
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
Formal power series
are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and
Generating_function
Result of repeatedly applying a mathematical function
Infinite compositions of analytic functions Flow (mathematics) Tetration Functional equation Half-exponential function while f (n) is taken for the nth
Iterated_function
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
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
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
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
Set of quantities in probability theory
number. The cumulants of the exponential distribution with rate parameter λ are κn = λ−n (n − 1)!. The cumulant generating function K(t), if it exists, is infinitely
Cumulant
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
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)
Continuous-time linear system with only negative real parts
left half of the complex plane). A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie
Exponential_stability
Henyey–Greenstein phase function The Mie phase function The von Mises distribution The wrapped normal distribution The wrapped exponential distribution The wrapped
List of probability distributions
List_of_probability_distributions
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
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
Function used in signal processing
the exponential window increases exponentially towards the center of the window and decreases exponentially in the second half. Since the exponential function
Window_function
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
Estimate of time taken for running an algorithm
Sometimes, exponential time is used to refer to algorithms that have T(n) = 2O(n), where the exponent is at most a linear function of n. This gives
Time_complexity
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
Probability distribution
versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special
Gamma_distribution
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
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
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
Probability distribution
{\displaystyle b=1} , the positive half-line is exactly an exponential distribution scaled by 1/2. The probability density function of the Laplace distribution
Laplace_distribution
Major unsolved problem in transcendental number theory
m1x1 +...+ mnxn = 0. This would be a positive solution to Tarski's exponential function problem. A related conjecture called the uniform real Schanuel's
Schanuel's_conjecture
Family of solutions to related differential equations
represent the Laplace distribution as an Exponential-scale mixture of normal distributions. The modified Bessel function of the second kind has also been called
Bessel_function
Arithmetic operation
integer Mathematics portal Double exponential function – Exponential function of an exponential function Exponential decay – Decrease in value at a rate
Exponentiation
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
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
Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Technique for designing differentially private algorithms
The exponential mechanism is a technique for designing differentially private algorithms. It was developed by Frank McSherry and Kunal Talwar in 2007
Exponential_mechanism
Type of statistical measure over subsets of a dataset
average Rolling hash Running total Savitzky–Golay filter Window function Zero lag exponential moving average Hydrologic Variability of the Cosumnes River
Moving_average
Product of numbers from 1 to n
for the gamma function at half-integers and the volumes of hyperspheres, and in counting binary trees and perfect matchings. Exponential factorial Just
Factorial
Compounding sum paid for the use of money
up interest in Wiktionary, the free dictionary. Credit card interest Exponential growth Fisher equation Interest Interest rate Rate of return Rate of
Compound_interest
Mathematical theorem
states that every real-valued function on the half-line [0, ∞) that is completely monotone is a mixture of exponential functions or in more abstract language
Bernstein's theorem on monotone functions
Bernstein's_theorem_on_monotone_functions
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)
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
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
Characteristic time in a system
represents the exponential decay constant and V is a function of time t V = V ( t ) . {\displaystyle V=V(t).} The right-hand side is the forcing function f(t) describing
Time_constant
Integral transform useful in probability theory, physics, and engineering
allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. Nachbin's theorem gives
Laplace_transform
The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between. The following is
Growth_function
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
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
as is eα for any non-zero algebraic α. Characterizations of the exponential function Transcendental number, including a proof that e is transcendental
Proof_that_e_is_irrational
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
Probability distribution
cumulative distribution function of an exponential distribution with rate α. Pareto distribution can be constructed by hierarchical exponential distributions.
Pareto_distribution
Feature observed in spectroscopy
include Lorentzian, Gaussian and Voigt functions, whose parameters are the line position, maximum height and half-width. Actual line shapes are determined
Spectral_line_shape
Inverse functions of sin, cos, tan, etc.
inverse function. sin ( ϕ ) = z ϕ = arcsin ( z ) {\displaystyle {\begin{aligned}\sin(\phi )&=z\\\phi &=\arcsin(z)\end{aligned}}} Using the exponential definition
Inverse trigonometric functions
Inverse_trigonometric_functions
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
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
Theorem in transcendental number theory
formulation—If α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over the algebraic numbers. This
Lindemann–Weierstrass_theorem
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
Operation in mathematical calculus
function does not have integrals that can be expressed in closed form involving only elementary functions, include rational and exponential functions
Integral
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
Time constant of an RC circuit
response Emphasis, preemphasis, deemphasis Exponential decay Filter (signal processing) and transfer function High-pass filter, low-pass filter, band-pass
RC_time_constant
Mathematical function for the probability a given outcome occurs in an experiment
cumulative distribution function of X = Finv(U) is F. For example, suppose we want to generate a random variable having an exponential distribution with parameter
Probability_distribution
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
Tool in multivariate statistical analysis
be written as a product of an exponential and a polynomial of degree p {\displaystyle p} . The modified Bessel function of a fractional order is given
Matérn_covariance_function
Functions such that f(–x) equals f(x) or –f(x)
be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and e x = cosh ( x )
Even_and_odd_functions
Inverse of a finite difference
function (or a constant if the Nørlund principal solution is assumed): Constant: ∑ x c = c x + C ( x ) {\displaystyle \sum _{x}c=cx+C(x)} Exponential:
Indefinite_sum
Instantaneous rate of change (mathematics)
trigonometric functions, p. 326 for the natural logarithm, pp. 338–339 for exponential with base e {\displaystyle e} , p. 343 for the exponential with base
Derivative
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)
secant function Integral of secant cubed Arclength Solid of revolution Shell integration Natural logarithm e (mathematical constant) Exponential function Hyperbolic
List_of_calculus_topics
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
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
Numeric quantity representing the center of a collection of numbers
distributions are in fact best described by their mean, including the exponential and Poisson distributions. The mean of a probability distribution is
Mean
Type of random mathematical object
with the exponential smoothing of intensity functions (FP-ESI) is an extension of the nonhomogeneous Poisson process. The intensity function of an FP-ESI
Poisson_point_process
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
Search algorithm finding the position of a target value within a sorted array
search problems in computational geometry and in numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree
Binary_search
formula[clarification needed] and is a special case of a general formula for the exponential function: e x / y = 1 + 2 x 2 y − x + x 2 6 y + x 2 10 y + x 2 14 y + x 2
List_of_representations_of_e
Function representing the number of primes less than or equal to a given number
the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then
Prime-counting_function
German mathematician
constructed the functional square root of the exponential function as a half-iteration of the exponential, i.e. a function φ such that φ(φ(z)) = exp(z). Kneser
Hellmuth_Kneser
Decomposition of periodic functions
coefficients (exponential form) of s {\displaystyle s} and r . {\displaystyle r.} When the real and imaginary parts of a complex function are decomposed
Fourier_series
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
Generalized mathematical function
the complex logarithm log(z) is the multivalued inverse of the exponential function ez : C → C×, with graph Γ log ( z ) = { ( z , w ) : w =
Multivalued_function
logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function" (PDF). U.P.B. Sci. Bull., Series
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Nonparametric measure of rank correlation
a moving window. Instead, the Hermite series based estimator uses an exponential weighting scheme to track time-varying Spearman's rank correlation from
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Variant Fourier transforms
sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely
Sine_and_cosine_transforms
Mathematical functions which are smooth but not analytic
In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its domain
Non-analytic_smooth_function
Middle quantile of a data set or probability distribution
is the minimum value for which the power law holds. The median of an exponential distribution with rate parameter λ is the natural logarithm of 2 divided
Median
Electrical circuit
amplifier's elements can be rearranged to produce exponential output, the logarithm's inverse function. Such an amplifier may be called an exponentiator
Log_amplifier
Electric circuit composed of resistors and capacitors
current}},} which can be rearranged according to the standard form for exponential decay: d V ( t ) d t = − 1 R C V ( t ) . {\displaystyle {\frac {\mathrm
RC_circuit
Particular case of the generalized extreme value distribution
restricted to the positive half-line. If X ∼ E x p o n e n t i a l ( 1 ) {\displaystyle X\sim \mathrm {Exponential} (1)} is an exponentially distributed variable
Gumbel_distribution
Probability distribution
normal distribution but finite moments. The relationship between the exponential distribution and the Laplace distribution allows for a simple method
Multivariate Laplace distribution
Multivariate_Laplace_distribution
Integrals not expressible in closed-form from elementary functions
{x^{c-1}}e^{-x}} (incomplete gamma function); for c = 0 , {\displaystyle c=0,} the antiderivative can be written in terms of the exponential integral; for c = 1 2
Nonelementary_integral
Generalization of the one-dimensional normal distribution to higher dimensions
simplifies to an exponential distribution with mean equal to two (rate equal to half). The complementary cumulative distribution function (ccdf) or the tail
Multivariate normal distribution
Multivariate_normal_distribution
Variable representing a random phenomenon
},\end{aligned}}} which is the cumulative distribution function (CDF) of an exponential distribution. Suppose X {\displaystyle X} is a random variable
Random_variable
Mathematical concept
More specifically the asymptotic expansion must include additional exponentially small terms relative to the usual algebraic terms included in a usual
Stokes_phenomenon
Analytic function in mathematics
statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article
Riemann_zeta_function
Probability distribution
probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. Roughly speaking, “heavy-tailed”
Heavy-tailed_distribution
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
Mathematical function
S2CID 16909853. Laforgia, Andrea; Natalini, Pierpaolo (2013). "Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities". Journal
Digamma_function
Methods of estimating the doubling time of an investment
calculations or when only a basic calculator is available. These rules apply to exponential growth and are therefore used for compound interest as opposed to simple
Rule_of_72
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
Boy/Male
American, Australian, British, English, German, Norse, Scandinavian, Teutonic
Half Dane; Half-danish
Male
German
Low German pet form of German Adolf, ALF means "noble wolf." Compare with other forms of Alf.
Boy/Male
Norse
Half son of Asgeir.
Male
Scandinavian
Scandinavian form of Old Norse Alfr, ALF means "elf." Compare with other forms of Alf.
Boy/Male
Arabic, Indian, Lebanese, Sanskrit, Swahili
Halo Around the Moon; Plough; Great; Dazzling; Sweetness
Male
Scandinavian
 Scandinavian form of Old Norse Ráðúlfr, RALF means "wise wolf." Compare with another form of Ralf.
Girl/Female
Muslim
Aureole, Halo around the Moon
Boy/Male
American, Anglo, Australian, British, Christian, English, French, German, Teutonic
Ingenious; From the Hall; Healthy Hero
Boy/Male
Christian & English(British/American/Australian)
From the Hall or Manor
Male
English
Short form of English Alfred, ALF means "elf counsel." Compare with other forms of Alf.
Male
English
Pet form of English Henry, HAL means "home-ruler."
Surname or Lastname
English, Scottish, Irish, German, and Scandinavian
English, Scottish, Irish, German, and Scandinavian : from Middle English hall (Old English heall), Middle High German halle, Old Norse hǫll all meaning ‘hall’ (a spacious residence), hence a topographic name for someone who lived in or near a hall or an occupational name for a servant employed at a hall. In some cases it may be a habitational name from places named with this word, which in some parts of Germany and Austria in the Middle Ages also denoted a salt mine. The English name has been established in Ireland since the Middle Ages, and, according to MacLysaght, has become numerous in Ulster since the 17th century.Hall is one of the commonest and most widely distributed of English surnames, bearing witness to the importance of the hall as a feature of the medieval village.
Girl/Female
Indian
Aureole, Halo around the Moon
Boy/Male
Latin
Half man half horse.
Male
English
 English surname transferred to forename use, derived from Old English heall "hall," hence "lives at the hall." Middle English name HALL means "to cover, conceal."
Girl/Female
Muslim
Lunar halo. Glory.
Female
Welsh
Welsh name HAF means "summer."
Male
German
Pet form of German Adolf, AHLF means "noble wolf."
Boy/Male
English Swedish Teutonic
Lives in the hall.
Girl/Female
African, Arabic, Hindu, Indian, Lebanese, Muslim, Pashtun, Sanskrit, Swahili
Halo Around the Moon; Plough; Girlfriend; Great; Dazzling; Glorious; Lunar Halo; Glory; Golden; Female Friend; Sweetness; Outline of Brightness Surrounding a Full Moon
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
Boy/Male
Arabic
Sea; Ocean
Male
Spanish
Spanish pet form of Portuguese/Spanish Rodrigo, RUY means "famous power."
Boy/Male
Hindu
Girl/Female
Indian
Girl/Female
Tamil
Luminous
Boy/Male
Indian
Boy/Male
Indian
Wolf
Boy/Male
Anglo, British, English
King Richard the Second Earl of Salisbury
Boy/Male
Indian, Punjabi, Sikh
Humble; Very Polite
Girl/Female
Bengali, Indian
Born in Good Place
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
a.
Imperfectly hatched; as, half-hatched eggs.
n.
The moon at the quarters, when half its disk appears illuminated.
a.
Consisting of a moiety, or half; as, a half bushel; a half hour; a half dollar; a half view.
a.
Consisting of some indefinite portion resembling a half; approximately a half, whether more or less; partial; imperfect; as, a half dream; half knowledge.
a.
Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function.
a.
Half-demented; half-witted.
n.
Half the length of a sword; close fight.
a.
Half-blooded.
a.
One of two equal parts into which anything may be divided, or considered as divided; -- sometimes followed by of; as, a half of an apple.
v. t. & i.
To form, or surround with, a halo; to encircle with, or as with, a halo.
a.
Half-blooded.
a.
Of half the whole or ordinary length, as a picture.
n.
One half of a shutter made in two parts for closing a porthole.
a.
Half-bred; imperfect.
n.
See Half deck, under Deck.
adv.
In an equal part or degree; in some pa/ appro/mating a half; partially; imperfectly; as, half-colored, half done, half-hearted, half persuaded, half conscious.
n.
The shape of a half-moon; a crescent.
a.
Half-filled.
a.
Done or happening at intervals of half an hour.