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Extension of the domain of an analytic function (mathematics)
mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds
Analytic_continuation
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Theorem in complex analysis
important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from
Monodromy_theorem
In many-body physics, the problem of analytic continuation is that of numerically extracting the spectral density of a Green function given its values
Numerical analytic continuation
Numerical_analytic_continuation
analysis, such as holomorphicity, the theory of algebraic curves, and analytic continuation. However, the numerical implementation is rather straightforward
Holomorphic Embedding Load-flow method
Holomorphic_Embedding_Load-flow_method
Analogue of a complex analytic space over a nonarchimedean field
p-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness. The basic rigid analytic object
Rigid_analytic_space
Infinite series that is not convergent
arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability
Divergent_series
Divergent series
zeta function is that it can be defined for other values of s by analytic continuation. One can then define the zeta-regularized sum of 1 + 2 + 3 + 4 +
1_+_2_+_3_+_4_+_⋯
Branch of mathematics studying functions of a complex variable
principle of analytic continuation which allows extending every real or complex analytic function in a unique way for getting a complex analytic function
Complex_analysis
Polygonal curve made from right triangles
{\displaystyle f(0)=1,} and monotonicity in both argument and modulus. An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in
Spiral_of_Theodorus
Global analytic functions arise naturally in considering the possible analytic continuations of an analytic function, since analytic continuations may have
Global_analytic_function
Study of space and shapes locally given by a convergent power series
function is conformal. Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining
Geometric_function_theory
American computer scientist (1922–1990)
titled "On Integral Equations, Their Solution by Iteration and Analytic Continuation". In 1952, he participated in Project Whirlwind. He joined the faculty
Alan_Perlis
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
Gamma_function
Class of mathematical function
z . {\displaystyle f(z)=\csc z={\frac {1}{\sin z}}.} By using analytic continuation to eliminate removable singularities, meromorphic functions can
Meromorphic_function
Meromorphic function on the complex plane
convergent on a half-plane, that may give rise to an L-function via analytic continuation, is called an L-series. Fundamental subclasses of L-functions were
L-function
Type of mathematical series
these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of z other
Bilateral hypergeometric series
Bilateral_hypergeometric_series
Mathematical theorem
the analytic continuation of lacunary power series. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function
Fabry_gap_theorem
the analytic continuation of lacunary power series. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function
Hadamard_gap_theorem
Point of interest for complex multi-valued functions
{\displaystyle z} . Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will
Branch_point
Generalization of the Riemann zeta function for algebraic number fields
zeta function: it can be defined as a Dirichlet series; it has an analytic continuation to a meromorphic function on the complex plane C with only a simple
Dedekind_zeta_function
Mathematics principle in complex analysis
of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper
Schwarz_reflection_principle
Type of Dirichlet series associated to number field extensions
), and for the rest of complex plane the function is defined by analytic continuation. Almost all prime ideals in a finite extension of a number field
Artin_L-function
Analytic function in mathematics
\mathrm {Re} (s)>1} , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications
Riemann_zeta_function
Mathematical techniques for summing divergent infinite series
}}f^{(2k-1)}(x)} where C is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan,
Ramanujan_summation
Type of mathematical function
{\displaystyle 1} . It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex
Dirichlet_L-function
Short "burst" or "envelope" of restricted wave action that travels as a unit
In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope
Wave_packet
Special mathematical function
z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator ks is understood as exp(s ln k)). The special
Polylogarithm
Mathematical functions which are smooth but not analytic
real analytic function is, at each point in its domain, the limit of a convergent power series in a neighbourhood of that point. All real analytic functions
Non-analytic_smooth_function
Method for estimating new data outside known data points
Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is
Extrapolation
Logarithm of a complex number
integration of 1 / z {\displaystyle 1/z} , or by the process of analytic continuation. There is no continuous complex logarithm function defined on all
Complex_logarithm
Mathematical behavior near singularities
explicit in complex analysis. In the process of analytic continuation, a function that is an analytic function F ( z ) {\displaystyle F(z)} in some open
Monodromy
Method in evaluating divergent integrals
them that are meromorphic functions of a complex parameter d, the analytic continuation of the number of spacetime dimensions. Dimensional regularization
Dimensional_regularization
Summation method for divergent series
{1}{1-z}}} which converges in the larger region Re(z) < 1, giving an analytic continuation of the original series. Considering instead the weak Borel transform
Borel_summation
Mathematical method for integrodifferential equations
strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane,
Wiener–Hopf_method
Weil cohomology theory for schemes X over a base field k
'propagation' notable in the case of the analytic continuation of complex analytic functions. (Cf. also the rigid analytic spaces introduced by John Tate, in
Crystalline_cohomology
Topics referred to by the same term
bounded analytic function can become Analytic continuation, a technique to extend the domain of definition of a given analytic function Analytic manifold
Analytic
Series of functions in mathematics
Additional integrals, convergent in overlapping sectors, allow analytic continuation in the complex plane for both types of series. This regularization
Asymptotic_expansion
Unproved conjecture in mathematics
conjectured that L ( E , s ) {\displaystyle L(E,s)} could be extended by analytic continuation to the whole complex plane.[citation needed] This conjecture was
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Theorem in complex analysis
Holomorphic function Meromorphic function Cauchy–Riemann equations Analytic continuation Series and singularities Power series Taylor series Laurent series
Cauchy's_integral_theorem
Mathematical conjecture about zeros of L-functions
number s such that Re s > 1 this series is absolutely convergent. By analytic continuation, this function can be extended to a meromorphic function on the
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Theorem of analytic continuations
on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the
Edge-of-the-wedge_theorem
the real part of s is greater than 3/2, and may be extended by analytic continuation to other values of s. Like the Riemann zeta function, whose value
Airy_zeta_function
Generalized mathematical function
originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f ( z ) {\displaystyle
Multivalued_function
Constants of the mathematical zeta function
function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1 {\displaystyle s=1} . The complex
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Association of one output to each input
complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost
Function_(mathematics)
Complex-differentiable (mathematical) function
Taylor series (is analytic). Holomorphic functions are the central objects of study in complex analysis. Though the term analytic function is often used
Holomorphic_function
Summability method in physics
{1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+\cdots } if this sum converges, and by analytic continuation elsewhere. In the case when an = n, the zeta function is the ordinary
Zeta_function_regularization
Mathematical concept
\pi _{v}} of local groups. The L-function is expected to have an analytic continuation as a meromorphic function of all complex s {\displaystyle s} , and
Automorphic_L-function
Type of mathematical functions
theory. A number of issues were clarified, in particular that of analytic continuation. Here a major difference is evident from the one-variable theory;
Function of several complex variables
Function_of_several_complex_variables
Class of integrals appearing in quantum field theory
can be written in terms of expressions which have a well-defined analytic continuation from integers n {\displaystyle n} to functions on C {\displaystyle
Loop_integral
Polish-French mathematician (1899–1983)
1923, he received a doctorate from the University of Paris on the analytic continuation of the Taylor series. Hadamard was his Ph.D. advisor. In 1924 Mandelbrojt
Szolem_Mandelbrojt
Numerical technique
mapped to complex numbers; more technically, this is actually an analytic continuation of the wave equation into complex coordinates, replacing propagating
Perfectly_matched_layer
Provides integral formulas for all derivatives of a holomorphic function
{a}{z}}\right)^{2}+\cdots }{z}},} it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series. In particular f
Cauchy's_integral_formula
One-dimensional complex manifold
Important examples of non-compact Riemann surfaces are provided by analytic continuation. If P ( x , y ) {\displaystyle P(x,y)} is any complex polynomial
Riemann_surface
Force resulting from the quantisation of a field
if the damping of large-frequency excitations corresponding to analytic continuation of the Riemann zeta function to s = 0 is assumed to make sense physically
Casimir_effect
Mathematical theorem about the real analytic Eisenstein series
n)\neq (0,0)}{y^{s} \over |m\tau +n|^{2s}}} for Re(s) > 1, and by analytic continuation for other values of the complex number s. γ is Euler–Mascheroni
Kronecker_limit_formula
23 mathematical problems stated in 1900
solutions of regular problems in the calculus of variations always necessarily analytic? The general problem of boundary values Proof of the existence of linear
Hilbert's_problems
Concept in complex analysis
respect to z at every point of U. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of U, and converges
Zeros_and_poles
Inverse of a finite difference
k ) . {\displaystyle \,\sum _{k=1}^{x}f(k).} In this case, the analytic continuation, F ( x ) {\displaystyle F(x)} , for the sum is a solution of ∇ −
Indefinite_sum
Topic in mathematical physics
powers of Euclidean space-time must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman
Axiomatic quantum field theory
Axiomatic_quantum_field_theory
Theorem about the range of an analytic function
and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. Little Picard Theorem: If
Picard_theorem
Unsolved problem in mathematics
of the complex plane, the Ramanujan L-function can be defined by analytic continuation of this series. Like other L-functions, the Ramanujan L-function
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Statement in complex analysis
Holomorphic function Meromorphic function Cauchy–Riemann equations Analytic continuation Series and singularities Power series Taylor series Laurent series
Hadamard factorization theorem
Hadamard_factorization_theorem
Power series with negative powers
{\displaystyle A} in which f ( z ) {\displaystyle f(z)} is holomorphic (analytic). The expansion for f ( z ) {\displaystyle f(z)} will then be valid anywhere
Laurent_series
Mathematical function that preserves angles
(orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply
Conformal_map
Theorem in complex analysis
0, it follows that g' (z)/g(z) has no singularities at zZ, and thus is analytic at zZ, which implies that the residue of f′(z)/f(z) at zZ is k. Let zP
Argument_principle
{O}}^{-s}} for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional
Zeta_function_(operator)
Divergent series
two formulas given above are not valid at zero however, but the analytic continuation is ζ ( s ) = 2 s π s − 1 sin ( π s 2 ) Γ ( 1 − s ) ζ ( 1
1_+_1_+_1_+_1_+_⋯
constructed some families of them for higher rank groups using analytic continuation, sometimes called the Stein complementary series. A.I. Shtern (2001)
Complementary series representation
Complementary_series_representation
Theorem on the equality of analytic functions
branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of R
Identity_theorem
which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Lejeune Dirichlet
History_of_calculus
Conjecture in topology
on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems. In 2013, John Pardon proved the three-dimensional
Hilbert–Smith_conjecture
Attribute of a mathematical function
such that f ( z ) − R / ( z − a ) {\displaystyle f(z)-R/(z-a)} has an analytic antiderivative in a punctured disk 0 < | z − a | < δ {\displaystyle 0<\vert
Residue_(complex_analysis)
Type of continuous map in topology
the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally
Covering_space
Function in harmonic analysis on groups
from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical functions have
Zonal_spherical_function
Quantum field theory at non-zero temperatures
Euclidean field theory, real-time observables can be retrieved by analytic continuation. The Feynman rules for gauge theories in the Euclidean time formalism
Thermal_quantum_field_theory
Seven mathematical problems with a US$1 million prize for each solution
{1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots .} Its analytical continuation has zeros at the negative even integers; that is, ζ ( s ) = 0 {\displaystyle
Millennium_Prize_Problems
Family of power series in mathematics
which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called
Generalized hypergeometric function
Generalized_hypergeometric_function
Number, approximately 3.14
Number Theory. Academic Press. pp. 262–265. Sondow, J. (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's
Pi
Number of times a curve wraps around a point in the plane
Holomorphic function Meromorphic function Cauchy–Riemann equations Analytic continuation Series and singularities Power series Taylor series Laurent series
Winding_number
Topics referred to by the same term
1965–1972 by Pink Floyd The Continuation, a 2009 album by Deathgaze Analytic continuation, in complex analysis Numerical continuation, to compute approximate
Continuation_(disambiguation)
Analytic function in mathematics
to define f on any open set containing the unit circle, making analytic continuation impossible as well. The argument above shows that certain series
Lacunary_function
Numerical technique for solving quantum Hamiltonians
"Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation". Physical Review E. 94 (2) 023303. arXiv:1507.01012. Bibcode:2016PhRvE
Exact_diagonalization
Property of scattering amplitudes
property is that the antiparticle scattering amplitudes are the analytic continuation of particle scattering amplitudes to negative energies. The interpretation
Crossing_(physics)
Degree of differentiability of a function or map
complex domain. Their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth
Smoothness
Theorem about zeros of holomorphic functions
usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler
Rouché's_theorem
Mathematical theorem in complex analysis
to unbounded domains. The Borel–Carathéodory theorem, which bounds an analytic function in terms of its real part. The Hadamard three-lines theorem, a
Maximum_modulus_principle
Mathematical trick using imaginary numbers to simplify certain formulas in physics
satisfies the Wightman axioms, all correlation functions admit an analytic continuation to Euclidean space. In addition, if a Euclidean QFT satisfies both
Wick_rotation
Infinite sum of monomials
such that no analytic continuation of the series can be defined at x. The power series expansion of the inverse function of an analytic function can be
Power_series
Theorem in complex analysis
Holomorphic function Meromorphic function Cauchy–Riemann equations Analytic continuation Series and singularities Power series Taylor series Laurent series
Weierstrass factorization theorem
Weierstrass_factorization_theorem
Mathematical theorem
domain bounded by the unit circle C 0 {\displaystyle C_{0}} and contains analytic arcs C i {\displaystyle C_{i}} and isolated points (the images of other
Riemann_mapping_theorem
Functions in mathematics
are real analogues to holomorphic functions. All harmonic functions are analytic, that is, they can be locally expressed as power series. This is a general
Harmonic_function
1859 mathematics paper by Bernhard Riemann
letter zeta (ζ) for a function previously mentioned by Euler The analytic continuation of this zeta function ζ(s) to all complex s ≠ 1 The entire function
On the Number of Primes Less Than a Given Magnitude
On_the_Number_of_Primes_Less_Than_a_Given_Magnitude
Theorem in complex analysis
theorem has several proofs. A standard analytical proof uses the fact that holomorphic functions are analytic. Proof If f {\displaystyle f} is an entire
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Mathematical finite graph-associated function
the Riemann hypothesis. The Ihara zeta function is defined as the analytic continuation of the infinite product ζ G ( u ) = ∏ p 1 1 − u L ( p ) , {\displaystyle
Ihara_zeta_function
Potential counterexample to the generalized Riemann hypothesis
d q ) {\textstyle \chi ~(\mathrm {mod} ~q)} is defined as the analytic continuation of the Dirichlet series: L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n − s L(s
Siegel_zero
Statement in complex analysis
known as the Schwarz–Pick theorem (after Georg Pick), characterizes the analytic automorphisms of the unit disc, i.e. bijective holomorphic mappings of
Schwarz_lemma
Mathematical function
the domain of a given function, for example by continuity or by analytic continuation. This means that it is not worthy to explicitly define the domain
Function_of_a_real_variable
Number with a real and an imaginary part
cosine and exponential, or, equivalently, by using the method of analytic continuation. The value of a trigonometric or hyperbolic function of a complex
Complex_number
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ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
ANALYTIC CONTINUATION
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