scholarly journals The problem of missing terms in term by term integration involving divergent integrals

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160567 ◽  
Author(s):  
Eric A. Galapon
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Finite Part ◽  
Stieltjes Transform ◽  
Complex Valued

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, ∫ 0 a f ( x ) ( ω + x ) − 1   d x , and the Stieltjes transform itself, ∫ 0 ∞ f ( x ) ( ω + x ) − 1   d x , is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f ( z )( ω + z ) −1 , with the divergent integrals arising from term by term integration interpreted as finite part integrals.

Complex Approximation and Simultaneous Interpolation on Closed Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 701-706 ◽  
Author(s):  
P. M. Gauthier ◽  
W. Hengartner
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Closed Subset ◽  
Finite Complex ◽  
Complex Approximation ◽  
Closed Sets ◽  
Limit Points ◽  
Simultaneous Interpolation ◽  
Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

2009 ◽  
Vol 05 (05) ◽  
pp. 805-830
Author(s):  
KAREN TAYLOR
Keyword(s):  
Analytic Continuation ◽  
Eisenstein Series ◽  
Complex Plane ◽  
Cusp Form ◽  
Vector Valued

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

scholarly journals Reflection Identities of Harmonic Sums of Weight Four

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 77 ◽  
Author(s):  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Linear Combination ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Bfkl Equation ◽  
Color Singlet ◽  
Pole Structure

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

scholarly journals Integro-differential equations with singular and hypersingular integrals

2019 ◽  
Vol 54 (4) ◽  
pp. 404-407 ◽  
Author(s):  
E. I. Zverovich ◽  
A. P. Shilin
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Singular Integrals ◽  
Closed Curve ◽  
Special Structure ◽  
Finite Part ◽  
Hypersingular Integrals ◽  
Hadamard Finite Part

Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.

scholarly journals On the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

2005 ◽  
Author(s):  
◽  
Vladimir Batchenko
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Complex Plane ◽  
Mean Value ◽  
Qualitative Behavior ◽  
Conditional Stability ◽  
One Dimensional ◽  
Jacobi Operators ◽  
The Mean ◽  
Complex Valued

In this thesis we characterize the spectrum of one-dimensional Schrödinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of Schrödinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the Schrödinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.

scholarly journals On Lucas-balancing zeta function

2018 ◽  
Vol 22 (1) ◽  
pp. 65-74
Author(s):  
Debismita Behera ◽  
Utkal Keshari Dutta ◽  
Prasanta Kumar Ray
Keyword(s):  
Zeta Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Riemann Zeta

In the present study a new modication of Riemann zeta function known as Lucas-balancing zeta function is introduced. The Lucas-balancing zeta function admits its analytic continuation over the whole complex plane except its poles. This series converges to a fixed rational number − ½ at negative odd integers. Further, in accordance to Dirichlet L-function, the analytic continuation of Lucas-balancing L-function is also discussed.

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