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Longer titles found: Functional equation (L-function) (view), Cauchy's functional equation (view)

searching for Functional equation 104 found (227 total)

alternate case: functional equation

Riemann–Siegel formula (853 words) [view diff] exact match in snippet view article find links to article

formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function

satisfy a number of analytic properties, including an important functional equation. The setting is in the generality of a connected quasi-split reductive

everywhere the sum of its Maclaurin series. The exponential satisfies the functional equation: exp ⁡ ( x + y ) = exp ⁡ ( x ) ⋅ exp ⁡ ( y ) . {\displaystyle

→ R {\displaystyle T=T_{w}:\mathbb {R} \to \mathbb {R} } to the functional equation T ( x ) = s ( x ) + w T ( 2 x ) . {\displaystyle T(x)=s(x)+wT(2x)

a meromorphic continuation to the complex plane and satisfies a functional equation with respect to s → n − s where n is the absolute dimension of X

mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters

meromorphic function of all complex s {\displaystyle s} , and satisfy a functional equation L ( s , π , r ) = ϵ ( s , π , r ) L ( 1 − s , π , r ∨ ) {\displaystyle

equations is expressed by Floquet theory. Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b   {\displaystyle f(z+\omega )=f(z)+az+b\

{\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore

satisfies the functional equation E ∗ ( z , s ) = E ∗ ( z , 1 − s )   {\displaystyle E^{*}(z,s)=E^{*}(z,1-s)\ } analogous to the functional equation for the

K. Schmidt treated general analytic number theory including the functional equation of the zeta function for function fields of arbitrary genus [Schm

theory Etherington, I. M. H. (1937), "Non-associate powers and a functional equation", Mathematical Gazette, 21 (242): 36–39, 153, doi:10.2307/3605743

Z(t)=e^{i\theta (t)}\zeta \left({\frac {1}{2}}+it\right).} It follows from the functional equation of the Riemann zeta function that the Z function is real for real

to f {\displaystyle f} and g {\displaystyle \mathbf {g} } . This functional equation is known as the Bellman equation, which can be solved for an exact

{\displaystyle F(x+1)-F(x)=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function

+ 0 + 0 + 0 + 0 + 0 + 0 + x16 + 0 + ⋯. This function satisfies a functional equation: F ( x ) = x − x 2 + x 4 − x 8 + ⋯ = x − [ ( x 2 ) − ( x 2 ) 2 +

-Pochhammer symbol. The q {\displaystyle q} -gamma function satisfies the functional equation Γ q ( x + 1 ) = 1 − q x 1 − q Γ q ( x ) = [ x ] q Γ q ( x ) {\displaystyle

the OEIS. The Dn are the unique monic polynomials satisfying the functional equation D n ( u + α u , α ) = u n + ( α u ) n , {\displaystyle D_{n}\left(u+{\frac

Max Deuring with his work (Koecher 1953) on Dirichlet series with functional equation where he introduced Koecher–Maass series. He qualified in 1954 at

appearing in the functional equation of its L-series, analogous to the way the conductor of a global field appears in the functional equation of its zeta function

and Automorphic L-Functions, AMS Colloquium Publications 58, 2010 Functional Equation Satisfied by Certain L-Functions, Compositio Math., vol. 37, 1978

function, a variant of the Riemann zeta function with a simpler functional equation Harish-Chandra's Ξ function, a special spherical function on a semisimple

turbulence. Another well-known result is the Feigenbaum-Cvitanović functional equation. P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay,

turns out to be an entire and R(s) = (2π)−sΓ(s)φ(s) satisfies the functional equation: R ( k − s ) = ( − 1 ) k / 2 R ( s ) ; {\displaystyle R(k-s)=(-1)^{k/2}R(s);}

JSTOR 1970610, MR 0219537 Doi, Koji; Naganuma, Hidehisa (1969), "On the functional equation of certain Dirichlet series", Inventiones Mathematicae, 9 (1): 1–14

determine the entire Lie flow, providing a solution to the translation functional equation f t ( f τ ( x ) ) = f t + τ ( x ) . {\displaystyle f_{t}(f_{\tau

non-zero ideals a of OK. The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation to the whole complex

function ρ {\displaystyle \rho } ; the problem reduces to solving the functional equation ρ ( x ) ρ ( y ) = ρ ( x cos ⁡ θ + y sin ⁡ θ ) ρ ( x sin ⁡ θ − y cos

A. Pfeiffer: On the Conformal Mapping of Curvilinear Angles. The Functional Equation φ [ f ( x ) ] = a 1 φ ( x ) {\displaystyle \varphi [f(x)]=a_{1}\varphi

Dec. 1972; vol.6, no.12, p. 3543-62 Corrigan, E; Hasslacher, B "Functional-Equation for Exponential Loop Integrals In Gauge Theories" Physics Letters

f(x).) Among the proofs and sketches of proofs: Two proofs of the functional equation of ζ(s) Proof sketch of the product representation of ξ(s) Proof

(z+1)=z\Gamma (z).\,} Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: log ⁡ Γ ( z + 1 ) = log

1287/opre.14.4.699, JSTOR 168733. Lawler, E. L.; Moore, J. M. (1969), "A functional equation and its application to resource allocation and sequencing problems"

Perrin-Riou's conjectural explicit reciprocity law related to the functional equation of p {\displaystyle p} -adic L-functions. Several contributions to

( s ) {\displaystyle \phi (s)} can be shown to obey the Kendall functional equation: 92  ϕ ( s ) = g [ s + λ − λ ϕ ( s ) ] {\displaystyle \phi (s)=g[s+\lambda

problem. Lanford gave the first proof that the Feigenbaum-Cvitanovic functional equation g ( x ) = T ( g ) ( x ) = ( 1 / λ ) g ( g ( λ x ) ) , g ( 0 ) = 1

be represented by equations that can be transformed into a single functional equation sometimes called a Bellman equation. This equation relates the benefits

I=I_{0}+\sum _{j=1}^{t-1}x_{j}-\sum _{j=1}^{t-1}d_{j}\geq 0} The functional equation representing minimal cost policy is: f t ( I ) = min x t ≥ 0 I +

}{\frac {x^{q^{i}}}{D_{i}}}.} The Carlitz exponential satisfies the functional equation e C ( T x ) = T e C ( x ) + ( e C ( x ) ) q = ( T + τ ) e C ( x )

three further developments. The first development is the study of functional equation and meromorphic continuation of the Hasse zeta function of a proper

of the main uses of the principle of permanence is to show that a functional equation that holds for the real numbers also holds for the complex numbers

work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which

Chandrasekharan, K.; Mandelbrojt, Szolem (1959). "On solutions of Riemann's functional equation". Bull. Amer. Math. Soc. 65 (6): 358–362. doi:10.1090/S0002-9904-1959-10372-4

of the fundamental properties of theta functions, including the functional equation and the Jacobi triple product formula, as well as many other results

function f ( x ) {\displaystyle f(x)} of a real variable satisfying the functional equation f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(x+y)=f(x)+f(y)}

that the true evolution of a physical system is a solution of the functional equation Hamilton's principle δ S δ q ( t ) = 0. {\displaystyle {\frac {\delta

be used to derive a variety of functional equations including the functional equation for the Riemann zeta function. One important such use of Poisson

∈ R {\textstyle \forall t\in \mathbb {R} } ). Moreover, via the functional equation, we can reflect these regions through s ↦ 1 − s {\textstyle s\mapsto

inversion theorem. Riemann used the Laplace transform to develop the functional equation of the Riemann zeta function, and this method[clarification needed]

Taiwanese J. Math., 7 (2003), no.1, 51-67 (with Vu Hoai An). On the functional equation P(f) = Q(g), Adv. Complex Anal. Appl., 3, Kluwer Acad. Publ., Boston

\infty !={\sqrt {2\pi }}} . From the logarithmic derivative of the functional equation, 2 ζ ′ ( 1 / 2 ) ζ ( 1 / 2 ) = log ⁡ ( 2 π ) + π cos ⁡ ( π / 4 )

invertible p-adic integers. The classical gamma function satisfies the functional equation Γ ( x + 1 ) = x Γ ( x ) {\displaystyle \Gamma (x+1)=x\Gamma (x)}

{\displaystyle b(1)} and b ( 2 ) {\displaystyle b(2)} , coupled with the functional equation b ( s + 2 ) = ( s + 1 ) 2 b ( s ) , {\displaystyle b(s+2)={\frac

approximation" schemes used in dynamic programming to solve Bellman's functional equation are based on fixed-point iterations in the space of the return function

splits a binary tree into two trees of smaller size. This yields the functional equation on the generating function B ( z ) = ∑ n = 0 ∞ B n z n : {\displaystyle

quaternions. Farkas gave conditions for the solvability of Schröder's functional equation in 1884. His results in hydrodynamics and thermodynamics were discussed

{x(1-y)}{1-xy}})+\operatorname {L} ({\frac {y(1-x)}{1-xy}})} . Abel's functional equation or five-term relation where L ⁡ ( z ) = π 6 [ Li 2 ⁡ ( z ) + 1 2

1. {\displaystyle \chi (-1)=-1.} This distinction appears in the functional equation of the Dirichlet L-function. The order of a character is its order

= x3 + ax+ b is smaller than 2. Because of the existence of the functional equation of the L-function of an elliptic curve, BSD allows us to calculate

1016/0001-8708(72)90018-7. Atkinson, F. V. (1963). "Some aspects of Baxter's functional equation". J. Math. Anal. Appl. 7: 1–30. doi:10.1016/0022-247X(63)90075-1

successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method. In fact, Dijkstra's

formula in his doctoral thesis to prove a more general version of the functional equation for the Riemann zeta function. This involves giving the zeta function

Mathematicae 41 (1991), no. 1, 85–93. Laha, R. G.; Lukacs, E. (1977). "On a functional equation which occurs in a characterization problem". Aequationes Mathematicae

value at s = 0 is taken to be the sum of the divergent series. The functional equation for this Padé zeta function is ∑ j = 0 n a j ζ R ( s − j ) = ∑ j

(s)>6} and by analytic continuation otherwise. It satisfies the functional equation L ( s ) Γ ( s ) ( 2 π ) s = L ( 12 − s ) Γ ( 12 − s ) ( 2 π ) 12

G. H. Hardy and J. E. Littlewood, when they deduced approximate functional equation for the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} and

_{h}:{\mathcal {H}}\to \mathbb {C} } satisfies for every g ∈ H the functional equation φ h ( g ) = ⟨ h , g ⟩ = ⟨ h ∣ g ⟩ {\displaystyle \varphi _{h}(g)=\langle

transition at J = h. When there is no external field, we can derive a functional equation that f ( β , 0 ) = f ( β ) {\displaystyle f(\beta ,0)=f(\beta )}

Combining this with the values Γ(⁠1/2⁠) = √π and Γ(1) = 1 and the functional equation zΓ(z) = Γ(z + 1) leads to V n ( R ) = 2 π n / 2 R n n Γ ( n 2 ) =

extend the zeta function to the entire complex plane and to prove its functional equation. For s = σ + it in the critical strip 0 < σ < 1, ζ ( s ) = s ∫ −

scaling function. This scaling function itself is a solution to a functional equation. In most situations it is useful to restrict ψ to be a continuous

properties, as can be seen from the properties of the continuous functional equation. For the last statement, let a k = a p k = Id ⁡ ( a ) ⋅ χ σ k ( a

3 + 4 + ⋯ = ⁠−+1/12⁠. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting

of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous

464 pages long. 1976 Eisenstein series. Langlands's proof of the functional equation for Eisenstein series was 337 pages long. 1983 Trichotomy theorem

{\displaystyle \exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z} by the functional equation and Euler's identity. For example, eiπ = e3iπ = −1 , so both iπ and

orders are order-isomorphic. It is an odd function, and satisfies the functional equation ?(x + 1) = ?(x) + 1; consequently x ↦ ?(x) − x is an odd periodic

translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number

{1}{w}}\right)=w^{-2n}p^{*}(w)p(w)=w^{-2n}f(w).} We can use this functional equation to prove that g, given by g = 1 | f ( w ) | 2 n | d w | 2 {\displaystyle

= p ( μ | I ) {\displaystyle g(\mu )=p(\mu |I)} must satisfy the functional equation: g ( μ ) = | ∂ μ ( 1 ) ∂ μ | g ( μ ( 1 ) ) = g ( μ + b ) {\displaystyle

n {\displaystyle n} -fold iteration. One then solves Schröder’s functional equation Φ b ( b w ) = s Φ b ( w ) {\displaystyle \Phi _{b}{\bigl (}b^{\,w}{\bigr

{\displaystyle y'={\tfrac {dy}{dx}}} denotes the derivative of y. Functional equation. The exponential function e x {\displaystyle e^{x}} is the unique

{\displaystyle f} is odd. Then Λ {\displaystyle \Lambda } satisfies the functional equation Λ ( s , f ) = ( − 1 ) ε Λ ( 1 − s , f ) . {\displaystyle \Lambda

Theory. New York: Macmillan. Laha, R. G.; Lukacs, E. (1977). "On a functional equation which occurs in a characterization problem". Aequationes Mathematicae

Möbius transformations g and h preserving the unit circle and disk. Functional equation. If |a|, |b| < 1 and Φ ( a , b ) = 1 2 π ∫ 0 2 π ( f ( e i θ ) −

\right)^{\frac {1}{2}}\vartheta (0;\tau )} was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform Γ

)=L(s,\psi )\prod _{p|k}\left(1-{\psi (p) \over p^{s}}\right)} The functional equation for the L-series is L ( 1 − s , ψ ) = k 1 s − 1 Γ ( s ) ( 2 π ) s

_{p}(s):={\frac {1}{1-p^{-s}}}} for each prime number p. In order to obtain a functional equation for ζ(s), one needs to multiply them with an additional term involving

can be obtained from the Legendre transformation of the Polchinski functional equation, derived by Joseph Polchinski in 1984. The concept of the effective

{1}{2}}s(s-1)\pi ^{-s/2}\Gamma ({\tfrac {1}{2}}s)\zeta (s)} satisfying the functional equation ξ ( s ) = ξ ( 1 − s ) , ∀ s ∈ C . {\displaystyle \xi (s)=\xi (1-s)

− n ) {\displaystyle \zeta (1-n)} is zero. By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:

}}+\beta V(c')} . (2) It can be proven that the solution to this functional equation, if it exists, is equivalent to the solution of (1). To prove its

\operatorname {eval} (B_{-}+a)^{p}=-p\zeta (1-p,a).} From the Riemann functional equation for Zeta function it follows that eval B + − p = eval ⁡ B + p + 1

given under series representations below. With a little help from a functional equation for the Hurwitz zeta function, the polylogarithm is consequently

1093/imrn/rnt047. S2CID 119121964. Langlands, Robert (1970). On the Functional Equation of the Artin L-Functions (PDF). pp. 1–287. Deligne, Pierre (1972)

^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)} (the functional equation of the Riemann zeta function) e − ζ ′ ( 0 ) = 2 π {\displaystyle

formulae, which turn out to be particular cases of the Hurwitz's functional equation. Speaking about the Malmsten's contribution into the theory of zeta-functions

(n)\equiv {\frac {\varphi (n)}{\,\Gamma (1+n)\,}}} and using the functional equation for the gamma function. The integral above is convergent for 0 <

3.5699 ⋯ {\displaystyle r^{*}=3.5699\cdots } , as we repeat the functional equation iteration f ( x ) ↦ − α f ( f ( − x / α ) ) {\displaystyle f(x)\mapsto

Geometry, Number Theory, and Combinatorics in an advanced level, Functional Equation, and Inequality. After the camp, an exam is given and 6 students

Belgrade: 19–28. arXiv preprint Ivić, Aleksandar (1995). "An approximate functional equation for a class of Dirichlet series". The Journal of Analysis. 3. Madras:

{\Gamma '}{\Gamma }}\left({\frac {1-s}{2}}\right)} of Riemann's functional equation, taken after applying Euler's reflection formula, these expressions

included determining the degree of the associated L-function, proving a functional equation, and determining the precise p-adic size of the roots (or poles,

τ f {\displaystyle \omega ={\color {orangered}\tau }f} Riemann's functional equation ζ ( s ) = 2 s π s − 1   sin ⁡ ( s π 2 )   Γ ( 1 − s )   ζ ( 1 − s