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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
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
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
exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy
List_of_exponential_topics
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
Arithmetic operation
tetration in Wiktionary, the free dictionary. Ackermann function Big O notation Double exponential function Hyperoperation Iterated logarithm Symmetric level-index
Tetration
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
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
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
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
Formal power series
are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and
Generating_function
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
Concept in statistics and wave theory
are half the maximum amplitude. Half width at half maximum (HWHM) is half of the FWHM if the function is symmetric. The term full duration at half maximum
Full_width_at_half_maximum
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
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
Base of natural logarithms
mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352 e
E_(mathematical_constant)
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
Complex exponential in terms of sine and cosine
fundamental relationship between the trigonometric functions and the complex exponential function. It is named after Leonhard Euler, who mentioned the
Euler's_formula
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
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
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
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
Probability distribution
versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special
Gamma_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
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
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
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
Arithmetic operation
system. Mathematics portal Double exponential function – Exponential function of an exponential function Exponential decay – Decrease in value at a rate
Exponentiation
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
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
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
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 do
Gamma_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
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)
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
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
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
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
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
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
Family of solutions to related differential equations
integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical
Bessel_function
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
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
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
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
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)
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
Inverse of a finite difference
represents an arbitrary 1-periodic function: Constant: ∑ c = c x + C ( x ) . {\displaystyle \sum c=cx+C(x).} Exponential: ∑ a x = a x − 1 a − 1 + C ( x )
Indefinite_sum
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
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
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 seminal
Riemann_zeta_function
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
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
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
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
Mathematical function
S2CID 16909853. Laforgia, Andrea; Natalini, Pierpaolo (2013). "Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities". Journal
Digamma_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
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
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
Growth function exhibiting a singularity at a finite time
functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic
Hyperbolic_growth
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
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
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
Extension of superfactorials to the complex numbers
\,\gamma } is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π {\displaystyle \Pi } denotes multiplication (capital pi notation)
Barnes_G-function
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
secant function Integral of secant cubed Arclength Solid of revolution Shell integration Natural logarithm e (mathematical constant) Exponential function Hyperbolic
List_of_calculus_topics
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
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
Heteroscedasticity-consistent standard errors Huber loss function Human subject research Hurst exponent Hyper-exponential distribution Hyper-Graeco-Latin square design
List_of_statistics_articles
Search algorithm used in sorted arrays
search problems in computational geometry and in numerous other fields. Exponential search extends binary search to unbounded lists. The binary search tree
Binary_search
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
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
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
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 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
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
Probability distribution
Tepper-García function, named after Mexican-born, German-Australian Astrophysicist Thor Tepper-García, is a combination of an exponential function and rational
Voigt_profile
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
Diminishing_returns
theorem Exponential function Beta function Gamma function Riemann zeta function Riemann hypothesis Generalized Riemann hypothesis Elliptic function Half-period
List of complex analysis topics
List_of_complex_analysis_topics
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
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
Mathematical theorem
{\displaystyle f(\zeta )=\int _{-A}^{A}F(x)e^{ix\zeta }\,dx} is an entire function of exponential type A {\displaystyle A} , meaning that there is a constant C {\displaystyle
Paley–Wiener_theorem
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
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
Probability distribution
cumulative distribution function of an exponential distribution with rate α. Pareto distribution can be constructed by hierarchical exponential distributions.
Pareto_distribution
Number of integers coprime to and less than n
( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n}
Euler's_totient_function
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
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
Continuous probability distribution
continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural
Logistic_distribution
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
Special function defined by an integral
}}\right)\end{aligned}}} Exponential integral Logarithmic integral Tanc function Tanhc function Sinhc function Coshc function Abramowitz & Stegun (1983)
Trigonometric_integral
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
Mathematical operation in calculus
{u'e^{u}}{e^{u}}}=u',} just as the logarithm of the exponential function of a function is just the original function. In summary, both derivatives and logarithms
Logarithmic_derivative
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
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
Open problem on 3x+1 and x/2 functions
many other ways to define a complex interpolating function, such as using the complex exponential instead of sine and cosine: f ( z ) = z 2 + 1 4 ( 2
Collatz_conjecture
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
Two kinds of probability distributions
{\displaystyle E_{1}} is the exponential function, the Mittag-Leffler distribution of order 1 {\displaystyle 1} is an exponential distribution. However, for
Mittag-Leffler_distribution
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HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
HALF EXPONENTIAL-FUNCTION
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