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Mathematical function
mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers
P-adic_exponential_function
Mathematical function, denoted exp(x) or e^x
Mittag-Leffler function, a generalization of the exponential function p-adic exponential function Padé table for exponential function – Padé approximation
Exponential_function
frequencies Plethystic exponential p-adic exponential function Power law Proof that e is irrational Proof that e is transcendental q-exponential Radioactive decay
List_of_exponential_topics
Branch of number theory
In mathematics, p-adic analysis is a branch of number theory that studies functions of p-adic numbers. Along with the more classical fields of real and
P-adic_analysis
specifically in p-adic analysis, the Artin–Hasse exponential, introduced by Emil Artin and Helmut Hasse in 1928, is the power series given by E p ( x ) = exp
Artin–Hasse_exponential
Number theory expression
Legendre's formula that the p-adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} . Legendre, A. M. (1830)
Legendre's_formula
Function in algebra
R), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the P-adic valuation, where P = (π). Its valuation
Valuation_(algebra)
Arithmetic operation
system. Mathematics portal Double exponential function – Exponential function of an exponential function Exponential decay – Decrease in value at a rate
Exponentiation
Type of function in mathematics
p {\displaystyle \mathbb {Q} _{p}} ∑ n = 0 ∞ a n x n {\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}} converges to an analytic function on the p-adic integers
Analytic_function
Mathematical function, inverse of an exponential function
(multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined
Logarithm
Open problem on 3x+1 and x/2 functions
{2}}\right)2^{k}.} The function Q is a 2-adic isometry. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost
Collatz_conjecture
Extension of the factorial function
gamma function Multivariate gamma function p-adic gamma function Pochhammer k-symbol Polygamma function q-gamma function Ramanujan's master theorem Spouge's
Gamma_function
Result in field theory about zeros of formal power series
valuation ring of the algebraic closure of K {\displaystyle K} . p-adic exponential function Straßmann, Reinhold (1928). "Über den Wertevorrat von Potenzreihen
Strassmann's_theorem
American mathematician (born 1945)
the degree as a rational function and for total degree of the associated L-function for a toric exponential sum, using the p-adic method developed by Bombieri
Steven_Sperber
octonions, sedenions, trigintaduonions etc.) p-adic function: a function whose domain is p-adic. Convex function: line segment between any two points on the
List_of_types_of_functions
Theorem in p-adic analysis
theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series is
Mahler's_theorem
Natural number
_{10}\left({\frac {d+1}{d}}\right)} . The tendency for real-world numbers to grow exponentially or logarithmically biases the distribution towards smaller leading digits
1
Product of numbers from 1 to n
the non-positive integers. In the p-adic numbers, it is not possible to continuously interpolate the factorial function directly, because the factorials
Factorial
Theorem in transcendental number theory
\mathbb {Q} } , such that | αi |p < 1/p for all i; then the p-adic exponentials expp(α1), . . . , expp(αn) are p-adic numbers that are algebraically independent
Lindemann–Weierstrass_theorem
Number divisible only by 1 and itself
_{p}=p^{-\nu _{p}(q)}} . Multiplying an integer by its p {\displaystyle p} -adic absolute value cancels out the factors of p {\displaystyle p}
Prime_number
Belgian mathematician
important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions. Deligne also focused on
Pierre_Deligne
Algebraic structure with addition, multiplication, and division
fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied
Field_(mathematics)
Group that is also a differentiable manifold with group operations that are smooth
\mathbb {Q} } , one can define a p-adic Lie group over the p-adic numbers, a topological group which is also an analytic p-adic manifold, such that the group
Lie_group
On generating functions from counting points on algebraic varieties over finite fields
on the ℓ-adic cohomology group Hi. The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from
Weil_conjectures
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
Fraction with denominator a power of two
a subsystem of the 2-adic numbers as well as of the reals, and can represent the fractional parts of 2-adic numbers. Functions from natural numbers to
Dyadic_rational
Recursive integer sequence
binomial coefficients, by Stirling's approximation for n!, or via generating functions. The only Catalan numbers Cn that are odd are those for which n = 2k −
Catalan_number
American mathematician (born 1943)
American mathematician, working in arithmetic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He is currently
Nick_Katz
Function whose domain is the positive integers
exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then
Arithmetic_function
Count of the possible partitions of a set
can be derived by expanding the exponential generating function using the Taylor series for the exponential function, and then collecting terms with the
Bell_number
Russian mathematician (1937–2008)
a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life
Anatoly_Karatsuba
On power series with rational coefficients that are algebraic functions
algebraic function with rational number coefficients. Through the theorem, it is readily demonstrable, for example, that the exponential function must be
Eisenstein's_theorem
Infinite sum of monomials
one of the most important examples of a power series, as are the exponential function formula e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ {\displaystyle
Power_series
Number, approximately 3.14
of π in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula: e i φ = cos
Pi
Number of ways to pair up n objects
is the value at zero of the n-th derivative of this function. The exponential generating function can be derived in a number of ways; for example, taking
Telephone number (mathematics)
Telephone_number_(mathematics)
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
Theorem on prime numbers
one to define the p-adic gamma function. Gauss proved that ∏ k = 1 gcd ( k , m ) = 1 m − 1 k ≡ { − 1 ( mod m ) if m = 4 , p α , 2 p α 1 ( mod m ) otherwise
Wilson's_theorem
Function which measures the "size" of elements in a field or integral domain
cases. Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (2nd ed.). New York: Springer-Verlag. p. 1. ISBN 978-0-387-96017-3. Retrieved
Absolute_value_(algebra)
Special functions of several complex variables
define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers. The Jacobi
Theta_function
Type of zeta function
zeta functions and mean-periodic functions in the space of smooth functions on the real line of not more than exponential growth. This correspondence is
Arithmetic_zeta_function
function. The most common examples are the complex numbers and all its subfields, and the valued fields, which include the p {\displaystyle p} -adic fields
Topological_ring
Topological group structure arising in Fourier analysis
/\mathbb {Z} } . The field Q p {\displaystyle \mathbb {Q} _{p}} of p-adic numbers under addition, with the usual p-adic topology. If G {\displaystyle
Locally_compact_abelian_group
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
{-\ln(1-z)}{1-z}},} where ln(z) is the natural logarithm. An exponential generating function is ∑ n = 1 ∞ z n n ! H n = e z ∑ k = 1 ∞ ( − 1 ) k − 1 k z
Harmonic_number
German mathematician (1898–1979)
the application of p-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Hasse was born in
Helmut_Hasse
On algebraic independence of logarithms
exponentials conjecture. Similarly, extending the result to algebraic independence but in the p-adic setting, and using the p-adic logarithm function
Baker's_theorem
Problem about mathematical number fields
and describe leading coefficients of Artin L-functions. In 2021, Dasgupta and Kakde announced a p-adic solution to finding the maximal abelian extension
Hilbert's_twelfth_problem
Branch of mathematics
arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers. A simple example that shows some
Complex_dynamics
Used to count, measure, and label
set of the p-adic numbers contains the rational numbers, but is not contained in the complex numbers. The elements of an algebraic function field over
Number
Infinite sum that is considered independently from any notion of convergence
seen as the (x)-adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p-adic integers are the p-adic completion of
Formal_power_series
Algebraic structure used in analysis
below. p-adic Lie groups are related to Lie algebras over the field Q p {\displaystyle \mathbb {Q} _{p}} of p-adic numbers as well as over the ring Z p {\displaystyle
Lie_algebra
Type of infinite structure
symbol for the exponential function by Wilkie's theorem. More generally, the complete theory of the real numbers with Pfaffian functions added. The last
O-minimal_theory
Mathematical concept named for Ernst Witt
standard p-adic integers. The main idea behind Witt vectors is that instead of using the standard p-adic expansion a = a 0 + a 1 p + a 2 p 2 + ⋯ {\displaystyle
Witt_vector
for simple theories Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable? The universality
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Conjecture on zeros of the zeta function
a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions. Several
Riemann_hypothesis
British mathematician and logician
various p-adic Poincaré series, and subsequently these methods have been applied to prove rationality of a wide range of generating functions in group
Angus_Macintyre
Product of the first "n" prime numbers
denoted by " p n # {\displaystyle p_{n}\#} ", is a function from natural numbers to natural numbers similar to the factorial function, but rather than
Primorial
Development of the mathematical function
While in modern terms, the logarithm function can be explained simply as the inverse of the exponential function or as the integral of 1/x, Napier worked
History_of_logarithms
Study of numbers that are not solutions of polynomials with rational coefficients
not related to the exponential function. The main results in transcendence theory tend to revolve around e and the logarithm function, which means that
Transcendental_number_theory
Particular kind of algebraic structure
algebras can also be defined over fields of p {\displaystyle p} -adic numbers. This is part of p {\displaystyle p} -adic analysis. The prototypical example of
Banach_algebra
is given by a locally integrable function. Harish-Chandra (1978, 1999) proved a similar theorem for semisimple p-adic groups. Harish-Chandra (1955, 1956)
Harish-Chandra's regularity theorem
Harish-Chandra's_regularity_theorem
Function in harmonic analysis on groups
operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake
Zonal_spherical_function
Mathematical sequence
{1}{k!}}\left({\frac {x}{1-x}}\right)^{k}} The n-th derivative of the function e 1 x {\displaystyle e^{\frac {1}{x}}} can be expressed with the Lah numbers
Lah_number
Special case of colimit in category theory
{\displaystyle i\leq j} . One obtains from this definition canonical functions ϕ j : A j → lim → A i {\displaystyle \phi _{j}\colon A_{j}\rightarrow
Direct_limit
Discrete analog of a derivative
multiplying these umbral basis exponentials. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols. Thus
Finite_difference
Meromorphic function on the complex plane
generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules. The statistics of the zero
L-function
Type of functional equation (mathematics)
ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric
Differential_equation
Branch of mathematics
the field of rational numbers, number fields, finite fields, function fields, and p-adic fields. A large part of singularity theory is devoted to the
Algebraic_geometry
Branch of mathematics
valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions. p-adic analysis
Mathematical_analysis
Polynomial equation whose integer solutions are sought
the sum of two or more unknowns, with coefficients, to a constant. An exponential Diophantine equation is one in which unknowns can appear in exponents
Diophantine_equation
Construction in representation theory
groups. Roger Howe found a version of the orbit method that applies to p-adic Lie groups. David Vogan proposed that the orbit method should serve as a
Orbit_method
Solved prime-number problem
fixed prime p {\displaystyle p} , define R ( n , p ) = ν p ( ( 2 n n ) ) {\displaystyle R(n,p)=\nu _{p}({\tbinom {2n}{n}})} to be the p-adic valuation of
Proof_of_Bertrand's_postulate
Type of number in combinatorial mathematics and statistics
binomial function: A m ( p , r ) ≡ r m p + r ( m p + r m ) = r m ( p − 1 ) + r ( m p + r − 1 m ) = r m ( m p + r − 1 m − 1 ) {\displaystyle A_{m}(p,r)\equiv
Fuss–Catalan_number
Four-dimensional number system
q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} =a+\mathbf {v} ,} the exponential is computed as exp ( q ) = ∑ n = 0 ∞ q n n ! = e a ( cos ‖ v ‖ +
Quaternion
Numbers obtained by adding the two previous ones
F_{1}=F^{\prime }(0)=1} , the exponential generating function of the Fibonacci numbers is given by the entire function F ( x ) = e φ x − e ψ x 5 {\displaystyle
Fibonacci_sequence
Concept in mathematics
module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of
Drinfeld_module
Algebraic object with an ordered structure
invariant total order Ordered exponential field – Ordered field with a function generalizing the exponential function Ordered group – Group with a compatible
Ordered_field
Type of prime number
factorial function; compare this with Wilson's theorem, which states that every prime p {\displaystyle p} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)!+1}
Wilson_prime
differentiation and p-adic fields. In 1979, Onneweer provided alternative definitions to the dyadic derivatives. Walsh function Haar wavelet Harmonic
Dyadic_derivative
Duality for locally compact abelian groups
numbers, and every finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally
Pontryagin_duality
Set of rules defining correctly structured programs
';'). Whether functions with the same identifier but different adicity are distinct is implementation-defined. If allowed, then a function CURVEAREA could
APL_syntax_and_symbols
Mathematics award
to the theory of Shimura varieties and the Riemann-Hilbert problem for p-adic varieties." 2021 Bhargav Bhatt – "For outstanding work in commutative algebra
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Theories in mathematical logic
The addition of further function symbols (e.g., the exponential function, the sine function) may change decidability. p-adic fields Ax & Kochen (1965)
List_of_first-order_theories
Rational number sequence
congruent modulo p − 1 to a particular a ≢ 1 mod (p − 1), and so can be extended to a continuous function ζp(s) for all p-adic integers Z p , {\displaystyle
Bernoulli_number
Algebraic structure with addition and multiplication
principal ideal (p) generated by a prime number p; it is called the ring of p-adic integers and is denoted Z p . {\displaystyle \mathbb {Z} _{p}.} The completion
Ring_(mathematics)
Mathematical sequences in combinatorics
formulae, may be found on the page for Stirling numbers and exponential generating functions. Another, infrequent notation is s 1 ( n , k ) {\displaystyle
Stirling_number
Two raised to an integer power
hardware, and the data is stored in one or more octets (23), double exponentials of two are common in computing. The first 21 of them are: Also see Fermat
Power_of_two
Numbers parameterizing ways to partition a set
_{r=1}^{k}{\frac {1}{1-rx}}={\frac {1}{x^{k+1}(1/x)_{k+1}}}} and have an exponential generating function given by ∑ n = k ∞ { n k } x n n ! = ( e x − 1 ) k k ! . {\displaystyle
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Number used to approximate the square root of 2
that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being
Pell_number
Number that is the product of factorials
any polynomial of n {\displaystyle n} , but more slowly than any exponential function of n {\displaystyle n} . More precisely, for every ε > 0 {\displaystyle
Jordan–Pólya_number
Prime such that p^2 divides 2^(p-1)-1
R.; Styer, R. (April 2004), "On px − qy = c and related three term exponential Diophantine equations with prime bases", Journal of Number Theory, 105
Wieferich_prime
Symbols for constants, special functions
matching number of a graph the p-adic valuation of a number Ξ {\displaystyle \Xi } represents: the original Riemann Xi function, i.e. Riemann's lower case
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Algorithms for calculating square roots
algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; one can, for example, construct
Square_root_algorithms
Algebraic ring that need not have additive negative elements
their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them. A generalization
Semiring
Concept in number theory
{\displaystyle {\left({\frac {b}{b-1}}\right)}^{k}\leq bk} , because of the exponential nature of ( b b − 1 ) k {\displaystyle {\left({\frac {b}{b-1}}\right)}^{k}}
Narcissistic_number
Kind of complex manifold
arXiv:math/0601337. doi:10.1215/S0012-7094-08-14111-0. S2CID 817920. - could be extended to complex tori p-adic Abelian Integrals: from Theory to Practice
Complex_torus
Topological group with compact topology
carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact
Compact_group
Number equal to the product of the sum and product of its digits
{\left({\frac {b}{b-1}}\right)}^{k}\leq k(b-1)^{2},} because of the exponential nature of ( b b − 1 ) k {\displaystyle {\left({\frac {b}{b-1}}\right)}^{k}}
Sum-product_number
Quadratic polynomial
Nevins and Thomas D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers[permanent dead link]" Wolf Jung : Homeomorphisms on Edges of the
Complex_quadratic_polynomial
The zeros of a linear recurrence relation mostly form a regularly repeating pattern
sequences with values in any field of characteristic zero. Its known proofs use p-adic analysis and are non-constructive. Let K {\displaystyle K} be a field of
Skolem–Mahler–Lech_theorem
Number of orderings allowing ties
definition of the exponential generating function and the right hand side is the function obtained from this summation. The form of this function corresponds
Ordered_Bell_number
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P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
P ADIC-EXPONENTIAL-FUNCTION
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