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Mathematical concept
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Meromorphic function on the complex plane
Dirichlet L-function Automorphic L-function Modularity theorem Artin conjecture Special values of L-functions Explicit formulae for L-functions Shimizu L-function
L-function
Index of lists with the same name
Riemann hypothesis. Selberg class S Explicit formulae for L-functions Trace formula A directory of all known zeta functions This set index article is a list
List_of_zeta_functions
Mathematical formula involving a given set of operations
the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed
Closed-form_expression
Mathematical relation consisting of a multi-variable function equal to zero
admit an explicit solution. The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely
Implicit_function
Topics referred to by the same term
by a rational exponent ) Explicit formulae (L-function), relations between sums over the complex number zeroes of an L-function and sums over prime powers
Explicit_formula
Formula whose values are the prime numbers
1080/00029890.2019.1530554. S2CID 127727922. Goodstein, R. L.; Wormell, C. P. (February 1967), "Formulae For Primes", The Mathematical Gazette, 51 (375): 35–38
Formula_for_primes
Generalizations of the Riemann zeta function
+ Re(si) > i for all i. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example
Multiple_zeta_function
Number, approximately 3.14
141592653589793238462643383279... It appears in many formulae across mathematics and physics. Some of these formulae are used as definitions of π to avoid relying
Pi
based on the explicit formulae of prime number theory, as they apply to Dirichlet L-functions, and other more general global L-functions. A single statement
Weil's_criterion
Constants of the mathematical zeta function
+ … . {\textstyle {\frac {1}{3^{2}}}+\ldots \,.} Explicit or numerically efficient formulae exist for ζ ( s ) {\displaystyle \zeta (s)} at integer arguments
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Extension of the factorial function
the complete gamma function for contrast.) An important category of exponentially decaying functions is that of Gaussian functions a e − ( x − b ) 2 c
Gamma_function
Equation in Fourier analysis
inversion formula Voronoi formula Discrete-time Fourier transform Explicit formulae for L-functions Riemann–Roch theorem § Arithmetic Riemann–Roch theorem Stein
Poisson_summation_formula
Equations for calculations of the Darcy friction factor
factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the
Darcy friction factor formulae
Darcy_friction_factor_formulae
Branch of mathematics
restrictive for functions of a complex variable than it is for functions of a real variable. Complex analysis studies holomorphic functions, the differentiable
Calculus
Association of one output to each input
Ross L. (1995). Calculus and Analytic Geometry (9th ed.). Addison-Wesley. ISBN 978-0-201-53174-9. The Wolfram Functions – website giving formulae and visualizations
Function_(mathematics)
Formal power series
possible for exponential generating functions; with an exponential generating function, it is an/n! that grows according to these asymptotic formulae. Generally
Generating_function
Polynomial equation of degree 3
numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by
Cubic_equation
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Coordinate transformation that preserves the form of Hamilton's equations
preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi
Canonical_transformation
Probability that random variable X is less than or equal to x
under the probability density function from negative infinity to x {\displaystyle x} . Cumulative distribution functions are also used to specify the distribution
Cumulative distribution function
Cumulative_distribution_function
Mathematical description of quantum state
measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities. Wave functions can be functions of variables other
Wave_function
Graph used in fluid dynamics
W. (1977). "Friction factor equation spans all fluid flow regimes". Chemical Engineering. 84 (24): 91–92. Friction loss Darcy friction factor formulae
Moody_chart
Theorem in mathematics
complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between
Inverse_function_theorem
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the
Theta_function
Mathematics concept
1016/0377-0427(93)90134-W. ISSN 0377-0427. Wolfram, D.A. (2024). "Inverse connection formulae for generalised Bessel polynomials". Bulletin of the Australian Mathematical
Bessel_polynomials
Probability distribution of the sum of random variables
For independent, continuous random variables with probability density functions (PDF) f , g {\displaystyle f,g} and cumulative distribution functions
Convolution of probability distributions
Convolution_of_probability_distributions
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
Tool for proving a logical formula
procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical
Method_of_analytic_tableaux
Transcendental single-variable function
{Cl} _{2}\left({\frac {2\pi }{3}}\right)} For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply
Clausen_function
Representation of a curve by a function of a parameter
expresses several quantities, such as the coordinates of a point, as functions of one or more variables called parameters. In the case of a single parameter
Parametric_equation
Logarithm to the base of the mathematical constant e
similar logarithmic functions near 1 for binary and decimal logarithms: log2(1 + x) and log10(1 + x). Similar inverse functions named "expm1", "expm"
Natural_logarithm
Polynomial sequence
explicitly as H n ( x ) = { n ! ∑ l = 0 n 2 ( − 1 ) n 2 − l ( 2 l ) ! ( n 2 − l ) ! ( 2 x ) 2 l for even n , n ! ∑ l = 0 n − 1 2 ( − 1 ) n − 1 2 − l
Hermite_polynomials
Conjecture on zeros of the zeta function
least conjecturaly satisfied by most functions usually called zeta functions or L-functions) than for functions defined by direct formula. Though it is
Riemann_hypothesis
Programming style in which control is passed explicitly
explicitly in the form of a continuation. This is contrasted with direct style, which is the usual style of programming. Gerald Jay Sussman and Guy L
Continuation-passing_style
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
Special functions
for instance. The more useful of the Goldberg, et al., formulae is the following: s Y l m ( θ , ϕ ) = ( − 1 ) l + m − s ( l + m ) ! ( l − m ) ! ( 2 l
Spin-weighted spherical harmonics
Spin-weighted_spherical_harmonics
Multivariate derivative (mathematics)
towards the 'steepest ascent' in some orientations. For differentiable functions where the formula for gradient holds, it can be shown to always transform
Gradient
Mathematical-logic system
that the lambda calculus treats functions "anonymously"; it does not give them explicit names. For example, the function s q u a r e _ s u m ( x , y )
Lambda_calculus
Logic formalism
formalized by a number of formulae, namely: Action precondition axioms, one for each action Successor state axioms, one for each fluent Axioms describing
Situation_calculus
Integral transform useful in probability theory, physics, and engineering
s-plane). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain
Laplace_transform
Polynomial sequence
are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in
Bernoulli_polynomials
Approximation of a function by its tangent line at a point
either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification
Linear_approximation
Representation method in chemistry
(though less frequently than skeletal formulae). For example, conformational structures look similar to skeletal formulae and are used to depict the approximate
Skeletal_formula
Criterion for the convergence of a series
ℓ {\displaystyle \ell } . Then we notice that for n ≥ n 0 + 1 {\displaystyle n\geq n_{0}+1} , | a n | > ℓ | a n − 1 | > ℓ 2 | a n − 2 | > . . . > ℓ n
Ratio_test
Relations between power sums and elementary symmetric functions
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between
Newton's_identities
Mathematical function with no sudden changes
where arguments and values of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between
Continuous_function
Formula in calculus
simplest form of the chain rule is for real-valued functions of one real variable. It states that if g is a function that is differentiable at a point
Chain_rule
Relationship between programs and proofs
as functions but it does not specify the class of functions relevant for the interpretation. If one takes lambda calculus for this class of function, then
Curry–Howard_correspondence
Instantaneous rate of change (mathematics)
derivatives of many functions from the derivatives of the basic functions: Constant rule If f {\displaystyle f} is a constant function, then for all x {\displaystyle
Derivative
Computation model defining an abstract machine
text; most of Chapter XIII "Computable functions" is on Turing machine proofs of computability of recursive functions, etc. Knuth, Donald E. (1973). The Art
Turing_machine
Mathematical approximation of a function
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the
Taylor_series
Differential operator in mathematics
maps Ck functions to Ck−2 functions for k ≥ 2. It is a linear operator Δ : Ck(Rn) → Ck−2(Rn), or more generally, an operator Δ : Ck(Ω) → Ck−2(Ω) for any open
Laplace_operator
methods for the numerical solution of the ordinary differential equation d y d t = f ( t , y ) . {\displaystyle {\frac {dy}{dt}}=f(t,y).} Explicit Runge–Kutta
List_of_Runge–Kutta_methods
Formula for the sum of an arithmetic function
Jose Javier Garcia (2024), Discrete Mellin Convolution and its Extensions, Perron Formula and Explicit Formulae, General Science Journal, ISSN:1916-5382
Perron's_formula
Method of mathematical integration
continuous functions, including elementary functions, for example polynomials. However, the graphs of other functions, for example the Dirichlet function, do
Lebesgue_integral
the truth functions of additional propositional connectives determine the truth values of complex propositional formulae in [0, 1]. Formulae that always
T-norm_fuzzy_logics
Differential calculus on function spaces
which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals
Calculus_of_variations
Notation of differential calculus
made explicit by putting its name as a subscript: if f is a function of a variable x, this is done by writing D x f {\displaystyle D_{x}f} for the first
Notation_for_differentiation
an MTL-algebra L if it is fully true under all evaluations in L, that is, if e(A) = 1 for all evaluations e in L. Some formulae (for instance, p → p)
Monoidal_t-norm_logic
Vector operator in vector calculus
sufficiently fast for |r| → ∞ can be decomposed uniquely into an irrotational part E(r) and a source-free part B(r). Moreover, these parts are explicitly determined
Divergence
Infinite sum
to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative
Series_(mathematics)
Product of numbers from 1 to n
0 ! = 1 {\displaystyle 0!=1} allows for the compact expression of many formulae, such as the exponential function, as a power series: e x = ∑ n = 0 ∞
Factorial
Typographical symbol (@)
labelled '@'.[citation needed] In chemical formulae, @ is used to denote trapped atoms or molecules. For instance, La@C60 means lanthanum inside a fullerene
At_sign
Mathematical conjecture about the Riemann zeta function
so-called Selberg trace formula bore a striking resemblance to the explicit formulae, which gave credibility to the Hilbert–Pólya conjecture. Hugh Montgomery
Hilbert–Pólya_conjecture
Type of mathematical expression
formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions
Polynomial
Branch of mathematical analysis
frequently necessary to be explicit about which definition is used. The corresponding derivative is calculated using Lagrange's rule for differential operators
Fractional_calculus
Set of quantities in probability theory
one sums only over the noncrossing partitions, then, by solving these formulae for the κ {\textstyle \kappa } in terms of the moments, one gets free cumulants
Cumulant
Family of implicit and explicit iterative methods
RUUNG-ə-KUUT-tah) are a family of explicit and implicit iterative methods, which include the Euler method, used in discretization for the approximate solutions
Runge–Kutta_methods
Formulation of classical mechanics
\ldots ,q_{N},t).} Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as Γ k
Hamilton–Jacobi_equation
Theorem: (cos x + i sin x)^n = cos nx + i sin nx
generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex
De_Moivre's_formula
Study of discrete mathematical structures
uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Topological
Discrete_mathematics
Model of computational complexity
size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, a prominent circuit class
Circuit_complexity
Fundamental result in the branch of mathematics known as character theory
motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those are built up from Dirichlet L-functions, or more general
Brauer's theorem on induced characters
Brauer's_theorem_on_induced_characters
Dimensionless parameter
voltage at the second resonator port. Explicit functions of the frequency-dependent inductive and capacitive couplings for pair of coupled resonant circuits
Coupling coefficient of resonators
Coupling_coefficient_of_resonators
Mathematical theorem
is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite
Selberg_trace_formula
Statement relating differentiable symmetries to conserved quantities
us see this explicitly. Let us say Q 1 [ L ] = ∂ μ f 1 μ {\displaystyle Q_{1}[{\mathcal {L}}]=\partial _{\mu }f_{1}^{\mu }} and Q 2 [ L ] = ∂ μ f 2 μ
Noether's_theorem
Operation in calculus
positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines
Integral
Open subset of the real–number line
zeta function of a fractal string, which is the geometric zeta function times the Riemann zeta function, may be used to write explicit formulae which
Fractal_string
Second-order partial differential equation
} which is a Fourier series for f. These trigonometric functions can themselves be expanded, using multiple angle formulae. Let the quantities u and v
Laplace's_equation
Branch of mathematics
measure. Many function spaces in analysis are defined using measures. The Lp spaces consist of functions whose powers are integrable, with functions identified
Mathematical_analysis
Logic that allows infinitely long proofs
a language with infinitely long formulae is being presented, it is not possible to write such formulae down explicitly. To get around this problem a number
Infinitary_logic
Mathematical notion of infinitesimal difference
substitution and integration by parts formulae for Stieltjes integral correspond, respectively, to the chain rule and product rule for the differential. Infinitesimal
Differential_(mathematics)
Count of permutations by cycles
(2017). "Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions". J. Integer Seq. 20 (3). arXiv:1610.09691
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Integral transform and linear operator
analytic functions, which has come to be known as the Riemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined
Hilbert_transform
Circulation density in a vector field
maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3
Curl_(mathematics)
Particular kind of exponential sum
the Riemann zeta function, primes in short intervals, primes in arithmetic progressions, the spectral theory of automorphic functions and related topics
Kloosterman_sum
trigonometric functions flourished in the Gupta period, especially due to Aryabhata (6th century AD), who discovered the sine function, cosine function, and versine
History_of_trigonometry
Type of derivative in mathematics
endogeneous variables are generally not explicit functions of the exogeneous variables, other than through the implicit function theorem, and the total derivative
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Roman god
meiri). Analogous Iranian formulae are to be found in an Avestic gāthā (Gathas). In other towns of ancient Latium the function of presiding over beginnings
Janus
Formula for the Legendre polynomials
{u^{2}}{2}}\right)} . These formulae are for the classical orthogonal polynomials. Similar formulae hold for many other sequences of orthogonal functions arising from
Rodrigues'_formula
Type of non-monotonic logic
logical formulae in W and all formulae in a default were originally assumed to be first-order logic formulae, but they can potentially be formulae in an
Default_logic
Foundational principle in quantum physics
space ℓ 2 ( Z / N Z ) {\displaystyle \ell ^{2}(\mathbb {Z} /N\mathbb {Z} )} of functions on the integers modulo N. This inequality has implications for signal
Uncertainty_principle
only a handful of forces result in formulae for u {\displaystyle u} in terms of known functions. The solution for φ {\displaystyle \varphi } can be expressed
Exact solutions of classical central-force problems
Exact_solutions_of_classical_central-force_problems
Deformation of the group algebra of a Coxeter group
an integral weight function is defined on W (i.e., a map L: W → Z with L(vw) = L(v) + L(w) for all v, w ∈ W with l(vw) = l(v) + l(w)), then a common specialization
Iwahori–Hecke_algebra
Process of mathematical modelling, performed on a computer
intense graphical displays, which transcended the world of numbers and formulae, sometimes also led to output that lacked a coordinate grid or omitted
Computer_simulation
Average uncertainty in variable's states
formula and very similar known formulae from statistical mechanics. In statistical thermodynamics the most general formula for the thermodynamic entropy S
Entropy_(information_theory)
inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This
Glossary_of_calculus
Mathematical operation
second-order Taylor polynomial for the function centered at x = a. For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors
Second_derivative
Parameters characterizing properties of our galaxy
\vert }_{R_{0}}d\cdot \cos ^{2}\left(l\right)-\Omega d\\\end{aligned}}} Using the sine and cosine half angle formulae, these velocities may be rewritten
Oort_constants
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EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
EXPLICIT FORMULAE-FOR-L-FUNCTIONS
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