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Special function defined by an integral
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number
Logarithmic_integral_function
Mathematical operation in calculus
mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′f where f′ is the derivative
Logarithmic_derivative
Special function defined by an integral
exponential integral Ei {\displaystyle \operatorname {Ei} } is a special function on the complex plane. It is defined as one particular definite integral of
Exponential_integral
Integrals not expressible in closed-form from elementary functions
(elliptic integral) 1 ln x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian integral) sin
Nonelementary_integral
a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x >
List of integrals of logarithmic functions
List_of_integrals_of_logarithmic_functions
Mathematical function, inverse of an exponential function
of x. A far better estimate of π(x) is given by the offset logarithmic integral function Li(x), defined by Li(x)=∫2x1ln(t)dt. The Riemann hypothesis
Logarithm
functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of Gaussian functions Gradshteyn, Ryzhik
Lists_of_integrals
Special function defined by an integral
mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions. The different sine integral definitions are Si
Trigonometric_integral
Functions of an angle
trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant
Trigonometric_functions
Indefinite integral
nonelementary integrals are the error function ∫e−x2dx, the Fresnel function ∫sinx2dx, the sine integral ∫sinxxdx, the logarithmic integral function ∫1logxdx
Antiderivative
Mathematical constant
mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von
Ramanujan–Soldner_constant
Mathematical function
\pi } denotes the prime counting function, Li {\displaystyle \operatorname {Li} } the logarithmic integral function with inverse Li − 1 {\displaystyle
Landau's_function
Extension of the factorial function
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic
Gamma_function
Logarithm to the base of the mathematical constant e
logarithmic identities Logarithm of a matrix Logarithmic coordinates of an element of a Lie group. Logarithmic differentiation Logarithmic integral function
Natural_logarithm
Large number used in number theory
{\displaystyle x} for which the prime-counting function π ( x ) {\displaystyle \pi (x)} exceeds the logarithmic integral function li ( x ) . {\displaystyle \operatorname
Skewes's_number
Function representing the number of primes less than or equal to a given number
equivalent statement is limx→∞π(x)li(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques
Prime-counting_function
{\displaystyle n} Radical function: The product of the distinct prime factors of a positive integer input. Logarithmic integral function: Integral of the reciprocal
List of mathematical functions
List_of_mathematical_functions
Mathematical function
antiderivatives; the integral of the Gaussian function is the error function: ∫exp(−x2)dx=π2erfx+C. Nonetheless, their improper integrals over the whole real
Gaussian_function
Family of solutions to related differential equations
where ψ(z) is the digamma function, the logarithmic derivative of the gamma function. There is also a corresponding integral formula (for Re(x) > 0):
Bessel_function
List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of area functions Partial derivative Disk integration
List_of_calculus_topics
Operation in calculus
below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are
Integral
Topics referred to by the same term
The term integral logarithm may stand for: Discrete logarithm in algebra, Logarithmic integral function in calculus. This disambiguation page lists articles
Integral_logarithm
Integration over a non-flat region in 3D space
integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns
Surface_integral
Mathematical concept
function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Conditions for switching order of integration in calculus
a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y} , then one can evaluate the double integral as an iterated integral: ∬
Fubini's_theorem
Conjecture on zeros of the zeta function
function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral
Riemann_hypothesis
Inverse functions of sin, cos, tan, etc.
argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above. All of these
Inverse trigonometric functions
Inverse_trigonometric_functions
Elementary functions and their finitely iterated integrals
extraction) and antiderivatives. The logarithmic function does not need to be explicitly included since it is the integral of 1 / x {\displaystyle 1/x} . It
Liouvillian_function
Characterization of how many integers are prime
Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li(x) (under the slightly different form of a series, which
Prime_number_theorem
Integral of sin(x)/x from 0 to infinity
as the sine integral, an antiderivative of the sinc function, is not an elementary function. In this case, the improper definite integral can be determined
Dirichlet_integral
Transcendental single-variable function
particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous
Clausen_function
list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integrals are antiderivative
List of integrals of exponential functions
List_of_integrals_of_exponential_functions
Method of evaluating certain integrals along paths in the complex plane
of evaluating certain integrals along paths in the complex plane. Contour integration is used to study complex-valued functions that are holomorphic in
Contour_integration
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Antiderivative of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative
Integral of the secant function
Integral_of_the_secant_function
Rules for computing derivatives of functions
if fi<n(x)>0 and dfidx exists. The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
Differentiation_rules
Mathematical function having a characteristic S-shaped curve or sigmoid curve
distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The
Sigmoid_function
Basic integral in elementary calculus
analysis, the Riemann integral is a rigorous definition of the integral of a function on an interval. It defines the integral by approximating the region
Riemann_integral
Definite integral of a scalar or vector field along a path
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear
Line_integral
Concept in mathematical analysis
a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in any of
Improper_integral
Measure for evaluating probabilistic forecasts
probability density function f : R n → R + {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} _{+}} . The multivariate logarithmic score is similar to the
Scoring_rule
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
List of formulae involving π – Uses of the constant List of integrals of logarithmic functions List of mathematical identities Lists of mathematics topics
List of logarithmic identities
List_of_logarithmic_identities
Mathematical transform that expresses a function of time as a function of frequency
mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various
Fourier_transform
Difference of two numbers divided by the logarithm of their quotient
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient
Logarithmic_mean
offset logarithmic integral function. li – logarithmic integral function or linearly independent. lim – limit of a sequence, or of a function. lim inf
List of mathematical abbreviations
List_of_mathematical_abbreviations
Topics referred to by the same term
domain (ccTLD) for Liechtenstein Li, the polylogarithm function Li, the logarithmic integral function <li></li>, indicating an item in an HTML list; see HTML
Li
Branch of mathematics
differential calculus and integral calculus. Differential calculus studies instantaneous rates of change and slopes of curves; integral calculus studies accumulation
Calculus
Concept of complex analysis
powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well
Residue_theorem
Analytic function in mathematics
study of the zeta function, following Riemann.) When Re(s) > 1, the function can be written as a converging summation or as an integral: ζ(s)=∑n=1∞1ns=1Γ(s)∫0∞xs−1ex−1dx
Riemann_zeta_function
Number, approximately 3.14
transform. This is the integral transform that takes a complex-valued integrable function f on the real line to the function defined as: f̂(ξ)=∫−∞∞f(x)e−2πixξdx
Pi
Formula for the derivative of a ratio of functions
Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms
Quotient_rule
Method of mathematical integration
the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and
Lebesgue_integral
Type of mathematical function
including the exponential integral (Ei) logarithmic integral (Li or li) and Fresnel integrals (S and C) the error function, erf(x)=2π∫0xe−t2dt, a fact
Elementary_function
Method of mathematical differentiation
calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative
Logarithmic_differentiation
S-shaped curve
"logarithmic" to "logistic" transition first noted by Pierre-François Verhulst, as noted above) and then reaching a maximal limit. A logistic function
Logistic_function
Differentiation under the integral sign formula
b(x)<\infty } and the integrands are functions dependent on x , {\displaystyle x,} the derivative of this integral is expressible as d d x ( ∫ a ( x )
Leibniz_integral_rule
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Integral using products instead of sums
systems of linear differential equations. The classical Riemann integral of a function f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } can be
Product_integral
Mathematical function with no sudden changes
curves and surfaces. Every continuous function f:[a,b]→ℝ is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the
Continuous_function
Method of calculating heat transfer in flow systems
In thermal engineering, the logarithmic mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow
Logarithmic mean temperature difference
Logarithmic_mean_temperature_difference
Theorem in complex analysis
poles of a meromorphic function to a contour integral of the function's logarithmic derivative. If f is a meromorphic function inside and on some closed
Argument_principle
completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Special function related to the dilogarithm
The inverse tangent integral is a special function, defined by: Ti 2 ( x ) = ∫ 0 x arctan t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac
Inverse_tangent_integral
Generalization of definite integrals to functions of multiple variables
multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables
Multiple_integral
Mathematical function having a characteristic "bell"-shaped curve
maximum at small x. Hence, the integral of a bell-shaped function is typically a sigmoid function. Bell shaped functions are also commonly symmetric. Many
Bell-shaped_function
functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions Residue theory Isometries
List of complex analysis topics
List_of_complex_analysis_topics
Method for evaluating indefinite integrals
the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller. The Risch algorithm is used to integrate elementary functions. These
Risch_algorithm
Mapping involving integration between function spaces
mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via integration
Integral_transform
Criterion for integration in terms of elementary functions
The function f:=1x, which exists in ℂ(x), does not have an antiderivative in ℂ(x). Its antiderivatives lnx+C do, however, exist in the logarithmic extension
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
Special mathematical function
polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the same notation without the subscript. Different polylogarithm functions in
Polylogarithm
Mathematical approximation of a function
tan x, sec x, ln sec x (the integral of tan), ln tan 1/2(1/2π + x) (the integral of sec, the inverse Gudermannian function), arcsec(√2 ex), and 2 arctan
Taylor_series
Integral over a 3-D domain
densities, or to calculate mass from a corresponding density function. Often the volume integral is represented in terms of a differential volume element
Volume_integral
Mathematical method in calculus
partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative
Integration_by_parts
Mathematical theorem, used in calculus
In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle
Integral_of_inverse_functions
2015-10-12. Ransford, Thomas (2010). "Computation of logarithmic capacity". Computational Methods and Function Theory. 10 (2): 555–578. doi:10.1007/BF03321780
List of mathematical constants
List_of_mathematical_constants
Mathematical relation consisting of a multi-variable function equal to zero
Contour line Isosurface Marginal rate of substitution Implicit function theorem Logarithmic differentiation Polygonizer Related rates Folium of Descartes
Implicit_function
Relationship between derivatives and integrals
states that for a continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Mathematical functions that quantify complexity
coordinates (e.g. 7 for the coordinates (3/7, 1/2)), but in a logarithmic scale. Height functions allow mathematicians to count objects, such as rational points
Height_function
Extension of superfactorials to the complex numbers
\exp(x)=e^{x}} is the exponential function, and Π {\displaystyle \Pi } denotes multiplication (capital pi notation). The integral representation, which may be
Barnes_G-function
Complex-differentiable (mathematical) function
v(x,y)} is holomorphic. Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes: ∮ γ f ( z )
Holomorphic_function
Solution of a confluent hypergeometric equation
Exponential integral and related functions such as the sine integral, logarithmic integral Hermite polynomials Incomplete gamma function Laguerre polynomials Parabolic
Confluent hypergeometric function
Confluent_hypergeometric_function
Identity expressing an integral as a sum
}}m\neq -1{\text{)}}\end{aligned}}} (also in the list of integrals of logarithmic functions). This reduces the power on the logarithm in the integrand
Sophomore's_dream
1 minus the cosine of an angle
the single angle (θ = 0, 2π, ...) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines. In fact, the
Versine
Relation of flow speed to wall distance
In fluid dynamics, the law of the wall (also known as the logarithmic law of the wall) states that the average velocity of a turbulent flow at a certain
Law_of_the_wall
Theorem in mathematics
theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous
Mean_value_theorem
Mathematical function
\operatorname {am} (u|m)} is a multivalued function (in u {\displaystyle u} ) with infinitely many logarithmic branch points (the branches differ by integer
Jacobi_elliptic_functions
logarithm Linear form in logarithms Linearithmic List of integrals of logarithmic functions Log canonical singularity Log-likelihood ratio Log-log graph
Index_of_logarithm_articles
Provides an asymptotic formula for counting the number of prime ideals of a number field
Theorem, a more precise estimate may be given in terms of the logarithmic integral function. The number of prime ideals of norm ≤ X is L i ( X ) + O K (
Landau_prime_ideal_theorem
Astronomical measure
magnitude system is logarithmic, the power law has logarithmic slope α + 1 {\displaystyle \alpha +1} . This is why a Schechter function with α = − 1 {\displaystyle
Luminosity function (astronomy)
Luminosity_function_(astronomy)
Integral transform useful in probability theory, physics, and engineering
(/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex
Laplace_transform
Term used in the theories of Riemann surfaces and algebraic curves
Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore
Differential of the first kind
Differential_of_the_first_kind
Technique in integral evaluation
the function f ( g ( x ) ) ⋅ g ′ ( x ) {\displaystyle f(g(x))\cdot g'(x)} is also integrable on [ a , b ] {\displaystyle [a,b]} . Hence the integrals ∫
Integration_by_substitution
Risk measure estimating the average loss in the worst tail of the distribution
incomplete gamma function, l i ( x ) = ∫ d x ln x {\displaystyle \mathrm {li} (x)=\int {\frac {dx}{\ln x}}} is the logarithmic integral function. If the loss
Expected_shortfall
Function that interpolates the factorial
we obtain uniqueness of this function, most often given by the Gamma function. The most common condition is the logarithmic convexity: this is the Bohr-Mollerup
Pseudogamma_function
Divergent sum of positive unit fractions
Hn=∑k=1n1k. These numbers grow very slowly, with logarithmic growth, as can be seen from the integral test. More precisely, by the Euler–Maclaurin formula
Harmonic_series_(mathematics)
Subset of real numbers that are greater than zero
measure on the real numbers under the logarithm: it is the length on the logarithmic scale. In fact, it is an invariant measure with respect to multiplication
Positive_real_numbers
Study of Galois symmetry groups of differential fields
and their indefinite integrals. Clearly, logarithmic functions, trigonometric functions, and their inverses are Liouville functions over C(x), and especially
Differential_Galois_theory
Mathematical integration method
Volkenborn integral is a method of integration for p-adic functions. Let : f : Z p → C p {\displaystyle f:\mathbb {Z} _{p}\to \mathbb {C} _{p}} be a function from
Volkenborn_integral
2D graphic with logarithmic scales on both axes
two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k
Log–log_plot
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LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
LOGARITHMIC INTEGRAL-FUNCTION
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