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Solution of a confluent hypergeometric equation
hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solution to Kummer's differential equation
Confluent hypergeometric function
Confluent_hypergeometric_function
Mathematical function
mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below
Kummer's_function
Function defined by a hypergeometric series
included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential
Hypergeometric_function
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Clausen function Complete Fermi–Dirac integral, an alternate form of the polylogarithm. Dilogarithm Incomplete Fermi–Dirac integral Kummer's function Riesz
List of mathematical functions
List_of_mathematical_functions
German mathematician (1810–1893)
divisors of binomial coefficients Kummer's function Kummer sum Kummer variety Kummer–Vandiver conjecture Kummer's transformation of series Ideal number
Ernst_Kummer
Mathematical functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied
Lemniscate_elliptic_functions
Result in number theory showing congruences involving Bernoulli numbers
(1964) used Kummer's congruences to define the p-adic zeta function. The simplest form of Kummer's congruence states that B h h ≡ B k k ( mod p ) whenever
Kummer's_congruence
Extension of the factorial function
gamma function (represented by Γ {\displaystyle \Gamma } , capital Greek letter gamma) is the most common extension of the factorial function to complex
Gamma_function
Special mathematical function
_{s}(-z)+\operatorname {Li} _{s}(z)=2^{1-s}\operatorname {Li} _{s}(z^{2}).} Kummer's function obeys a very similar duplication formula. This is a special case of
Polylogarithm
Family of power series in mathematics
the Theory of Bessel functions" (1966), Section 10.7, Equation (10) "DLMF: §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent
Generalized hypergeometric function
Generalized_hypergeometric_function
Difference between logarithm and harmonic series
{k}{k+3^{2}}}+\cdots \end{aligned}}} From the Malmsten–Kummer expansion for the logarithm of the gamma function we get: γ = log π − 4 log ( Γ ( 3 4 ) ) + 4
Euler's_constant
Types of special mathematical functions
{z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s
Incomplete_gamma_function
Identity obeyed by many special functions related to the gamma function
identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values;
Multiplication_theorem
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Sequence of differential equation solutions
{1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L n ( α ) ( x ) := (
Laguerre_polynomials
Equations of motion for viscous fluids
equations in Cartesian coordinate can be given with the help of the Kummer's functions with quadratic arguments. For the compressible Navier–Stokes equations
Navier–Stokes_equations
Collective term for blood tests used to check the function of the thyroid
Thyroid function tests (TFTs) is a collective term for blood tests used to check the function of the thyroid. TFTs may be requested if a patient is thought
Thyroid_function_tests
Kummer sums was made by J. W. S. Cassels, again building on previous ideas of Tomio Kubota. This was a product formula in terms of elliptic functions
Kummer_sum
p-adic L-function (Kubota & Leopoldt 1964)—is via the p-adic interpolation of special values of L-functions. For example, Kubota–Leopoldt used Kummer's congruences
P-adic_L-function
In mathematics, a solution to a modified form of the confluent hypergeometric equation
solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by M κ , μ ( z ) = exp
Whittaker_function
Function for Heun's differential equation
hypergeometric differential equations obtained by Kummer.[citation needed] The symmetries fixing the local Heun function form a group of order 24 isomorphic to the
Heun_function
Karush–Kuhn–Tucker conditions, above Kuiper belt – Gerard Kuiper Kummer's function, Kummer surface – Ernst Kummer Kuramoto model – Yoshiki Kuramoto Lagrangian mechanics
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Mathematical function
In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
Riesz_function
Irreducible nodal surface
In algebraic geometry, a Kummer quartic surface, first studied by Ernst Kummer (1864), is an irreducible nodal surface of degree 4 in P 3 {\displaystyle
Kummer_surface
Special function defined by an integral
is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument
Exponential_integral
Natural number
Adrien-Marie Legendre to express the asymptotic behavior of the prime-counting function. The Weil's conjecture on Tamagawa numbers states that the Tamagawa number
1
Product of numbers from 1 to n
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and
Factorial
abelian variety is called a Kummer surface. Shimura, Goro (1998), Abelian varieties with complex multiplication and modular functions, Princeton Mathematical
Kummer_variety
Type of Dirichlet series associated to number field extensions
zero of Artin L-function at s = 0 {\displaystyle s=0} is one. He predicted existence of Stark units whose roots should generate Kummer extensions of K
Artin_L-function
Type of discrete orthogonal polynomials
Pearce; L. Reichel; K.C. Richards (1998). "Gram Polynomials and the Kummer Function". Journal of Approximation Theory. 94: 128–143. doi:10.1006/jath.1998
Discrete Chebyshev polynomials
Discrete_Chebyshev_polynomials
Special function defined by an integral
Fresnel integrals S(x) and C(x), and their auxiliary functions F(x) and G(x) are transcendental functions named after Augustin-Jean Fresnel that are used in
Fresnel_integral
American actress
and then another, Kummer's narration described how each spotlighted organ functioned. On August 3, 1946, in Sheboygan, Wisconsin, Kummer married Raymond
Eloise_Kummer
Special functions in mathematics
are poles), but which are not generally solvable in terms of elementary functions. They were discovered by Émile Picard (1889), Paul Painlevé (1900, 1902)
Painlevé_transcendents
Part of mathematical queueing theory
Laplace transform can be expressed in terms of Kummer's function. The stationary probability mass function is a Poisson distribution π k = ( λ / μ ) k e
M/M/∞_queue
Number
zero function (or zero map) on a domain D. This is the constant function with 0 as its only possible output value, that is, it is the function f defined
0
Set of four hypergeometric series
Ξ2, which generalize Kummer's confluent hypergeometric function 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable
Appell_series
Polynomial sequence
)},\end{aligned}}} where 1F1(a, b; z) = M(a, b; z) is Kummer's confluent hypergeometric function. H e 2 n ( x ) = ( − 1 ) n ( 2 n − 1 ) ! ! 1 F 1 ( − n
Hermite_polynomials
Maširević, Dragana; Pogány, Tibor K. (2017). "Series of Bessel and Kummer-Type Functions". Lecture Notes in Mathematics. Cham: Springer International Publishing
Kapteyn_series
Failure of the soft palate to prevent airflow through the nose during speech
method of measuring the acoustic correlates of resonance and velopharyngeal function through a computer-based instrument. Nasometry testing gives the speech
Velopharyngeal_insufficiency
Physical interaction in post-classical physics
where M {\displaystyle M} is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we have used the integral
Static forces and virtual-particle exchange
Static_forces_and_virtual-particle_exchange
Number of subsets of a given size
previous generating function after the substitution x → x y {\displaystyle x\to xy} . A symmetric exponential bivariate generating function of the binomial
Binomial_coefficient
Mucosa that serves to moisten and protect the airways
laryngopharynx, where instead the epithelium is stratified squamous. It also functions as a barrier to potential pathogens and foreign particles, preventing
Respiratory_epithelium
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Rational number sequence
umbra Bell number Euler number Genocchi number Kummer's congruences Poly-Bernoulli number Hurwitz zeta function Euler summation Stirling polynomial Sums of
Bernoulli_number
Generalization of the hypergeometric differential equation
The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is P
Riemann's differential equation
Riemann's_differential_equation
variables that generalize Kummer's confluent hypergeometric series 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable.
Humbert_series
Study of computable functions and Turing degrees
computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study
Computability_theory
German mathematician (1805–1859)
series and was one of the first to give the modern formal definition of a function. In mathematical physics, he studied potential theory, boundary-value problems
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Grasshoff, Christian Kohlund, Hanns Zischler Drama Geschlecht: weiblich Dirk Kummer Ulrike Krumbiegel, Adriana Altaras, Inga Busch [de], Sabine Orléans [de]
List of German films of the 2000s
List_of_German_films_of_the_2000s
Probability distribution
characteristic function is the Fourier transform of the probability density function. The characteristic function of the beta distribution is Kummer's confluent
Beta_distribution
Product of prime numbers, plus one
infinitely many prime numbers. A Euclid number of the second kind (also called Kummer number) is an integer of the form En = pn # − 1, where pn # is the nth primorial
Euclid_number
Branch of number theory
algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization
Algebraic_number_theory
Sum in algebraic number theory
thus can be used to calculate certain zeta functions. Quadratic Gauss sum Elliptic Gauss sum Jacobi sum Kummer sum Kloosterman sum Gaussian period Hasse–Davenport
Gauss_sum
Criterion for the convergence of a series
negligible compared to the other terms, ρ Kummer {\displaystyle \rho _{\text{Kummer}}} may be written: ρ Kummer = n ln ( n ) ( a n a n + 1 − 1 ) − ln
Ratio_test
Largest integer that divides given integers
gcd(a/d, b/d) = 1. The GCD is a commutative function: gcd(a, b) = gcd(b, a). The GCD is an associative function: gcd(a, gcd(b, c)) = gcd(gcd(a, b), c). Thus
Greatest_common_divisor
normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Probability distribution
, b , z ) {\displaystyle M(a,b,z)} is Kummer's confluent hypergeometric function. The characteristic function is given by: φ ( t ; k ) = M ( k 2 , 1
Chi_distribution
{\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.} If we define the function E ( x ) = ∫ 0 π / 2 1 − x sin 2 θ d θ , {\displaystyle E(x)=\int _{0}^{\pi
Perimeter_of_an_ellipse
German mathematician (1843–1921)
Sobieszów, Poland). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter
Hermann_Schwarz
Medical diagnostic method
Patsalis PC, Urban A, Kummer M, Vasileva S, Stachon A, Hering S, Dietrich JW (12 November 2020). "Abnormal thyroid function is common in takotsubo syndrome
Thyroid's_secretory_capacity
Theory in number theory
Iwasawa theory, Tate's thesis, Kummer theory, ramification theory) Analytic number theory (analytic theory of L-functions, probabilistic number theory,
Anabelian_geometry
Formula in number theory
invariants of an algebraic number field to a special value of its Dedekind zeta function. We start with the following data: K is a number field. [K : Q] = n = r1
Class_number_formula
Numerical method in computational electromagnetics
methods such as Kummer's transformation and Ewald summation is often used to accelerate the computation of the periodic Green's function. Analytical regularization
Method of moments (electromagnetics)
Method_of_moments_(electromagnetics)
German mathematician (1823–1891)
the work of man"). Kronecker was a student and life-long friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Liegnitz, Prussia (now
Leopold_Kronecker
Gives information about the Galois module structure of class groups of cyclotomic fields
class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890). Let
Stickelberger's_theorem
Problem about mathematical number fields
the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field
Hilbert's_twelfth_problem
Branch of Galois theory in mathematics
branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. Artin
Artin–Schreier_theory
Branch of pure mathematics
mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical
Number_theory
Generalization of covers
Branched coverings are easily constructed as Kummer extensions, i.e. as algebraic extension of the function field. The hyperelliptic curves are prototypic
Branched_covering
theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli numbers). In its early days in the
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Result due to Kummer on cyclic extensions of fields that leads to Kummer theory
cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of
Hilbert's_Theorem_90
Macrophage cell of the skin
"Langerin/CD207 Sheds Light on Formation of Birbeck Granules and Their Possible Function in Langerhans Cells". Immunologic Research. 28 (2): 93–107. doi:10.1385/IR:28:2:93
Langerhans_cell
Sum type in number theory
interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character
Quadratic_Gauss_sum
Polish-Austrian mathematician
Vienna, Austria. The Mertens function M(x) is the sum function for the Möbius function, in the theory of arithmetic functions. The Mertens conjecture concerning
Franz_Mertens
Mathematical criterion about whether a series converges
)\to \mathbb {R} _{+}} be a non-negative and monotonically decreasing function such that f ( n ) = a n {\displaystyle f(n)=a_{n}} . If ∫ 1 ∞ f ( x ) d
Convergence_tests
Identity between theta functions of Riemann surfaces
theta functions of Riemann surfaces introduced by John Fay. Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of
Fay's_trisecant_identity
17th-century conjecture proved by Andrew Wiles in 1994
Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work
Fermat's_Last_Theorem
Galois extension whose Galois group is abelian
provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields. There
Abelian_extension
Proposed reconstructed word list for the Proto-Indo-European language
Monier Williams, p. 241. Lühr, Rosemarie (2014). "Spinne am Morgen bringt Kummer und Sorgen". Denkströme. 13. Haruyuki Saito. Das Partizipium Präteriti im
Indo-European_vocabulary
Type of cellular receptor that facilitates taste
nasal respiratory epithelium and more. In most of the organs, the receptor function is unclear. The sweet taste receptor found in the gut and in the pancreas
Taste_receptor
American mathematician (1878–1966)
relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. He was President of the American
Arthur_Byron_Coble
Number divisible only by 1 and itself
the zeros of the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} are located. This function is an analytic function on the complex numbers. For
Prime_number
German mathematician (1854–1925)
D) in 1878 at the University of Berlin, where his supervisors were Ernst Kummer and Karl Weierstrass. He contributed to the solution of the prime number
Hans Carl Friedrich von Mangoldt
Hans_Carl_Friedrich_von_Mangoldt
Capital and largest city of Austria
destroyed, so they were left standing. They now stand empty and serve no function, though various other such towers in the city were repurposed, such as
Vienna
Species of baboon
Swedell 2002:b "Papio hamadryas (hamadryas baboon)". Animal Diversity Web. Kummer, 1968 Swedell 2006 Stammbach, 1987 Schreier and Swedell 2009 Städele, Pines
Hamadryas_baboon
Two-dimensional conformal field theory
spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically. Liouville theory describes
Liouville_field_theory
small values of d > 1. The interest in nineteenth century geometry in the Kummer surface came in part from the way a quartic surface represented a quotient
Equations defining abelian varieties
Equations_defining_abelian_varieties
Algebraic structure with addition, multiplication, and division
complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields
Field_(mathematics)
Human protein and coding gene
protein 1 (PYPAF1). NLRP3 is a component of the innate immune system that functions as a pattern recognition receptor (PRR) – a cytosolic sensor that responds
NLRP3
On finite sums of products of three binomial coefficients, and a hypergeometric sum
1⁄2a − b − c) > 0. As c tends to −∞ it reduces to Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the
Dixon's_identity
German mathematician (1831–1916)
Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. (Thus Dedekind can be said to have been Kummer's most
Richard_Dedekind
Mental disorder
interesting enough. Pathological lying shows a complex relationship with brain function. Compulsive lying has been reported in multiple neurological disorders
Pathological_lying
Mathematical theorem
the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided
Artin_reciprocity
Algebraic structure with addition and multiplication
may also be non-numerical objects such as polynomials, square matrices, functions, and power series. More formally, a ring is a set that is endowed with
Ring_(mathematics)
vanishing of an L-function at s = 0 is one, Stark's refined conjecture predicts the existence of Stark units, whose roots generate Kummer extensions of K
Stark_conjectures
fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According to Weierstrass in his paper,
List_of_conjectures
Mathematical law, a generalization of quadratic reciprocity
{p^{n}}{q}}\right\}} where n is some integer prime to l such that pn is principal. The Kummer reciprocity law states that { p q } = { q p } {\displaystyle \left\{{\frac
Reciprocity_law
American mathematician
in September 1871. In Berlin he attended lectures of Weierstrass, Ernst Kummer, Helmholtz and Dove. In Leipzig he heard Karl Neumann, Bruhns, Mayer, Van
William_Edward_Story
Probability distribution in physics
\,{\tfrac {1}{2}}\chi ^{2})}}} where M(·,·,·) is the Kummer's confluent hypergeometric function.[circular reference] The variance is: σ 2 = c 2 ( χ 2
ARGUS_distribution
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