Contour integrals of analytic functions given on a grid in the complex plane

2020 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Complex Plane ◽  
Line Segment ◽  
Analytic Functions ◽  
Trapezoidal Rule ◽  
Contour Integrals ◽  
Grid Points ◽  
Maclaurin Formula

Abstract Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a $5\times 5$ ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from $2$nd to $26$th order.

scholarly journals A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130571 ◽  
Author(s):  
Mohsin Javed ◽  
Lloyd N. Trefethen
Keyword(s):  
Error Bound ◽  
Quadrature Formula ◽  
Analytic Functions ◽  
Trapezoidal Rule ◽  
Contour Integral ◽  
Periodic Functions ◽  
Midpoint Rule ◽  
Geometric Convergence ◽  
Maclaurin Formula

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another. Similar results are also given for the midpoint rule.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

scholarly journals The arc-length variation of analytic capacity and a conformal geometry

1992 ◽  
Vol 125 ◽  
pp. 151-216
Author(s):  
Takafumi Murai
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Conformal Geometry ◽  
Supremum Norm ◽  
Length Variation ◽  
Analytic Capacity ◽  
Arc Length ◽  
Bounded Analytic Functions ◽  
Extended Complex

For a domain Ω in the extended complex plane C ∪{∞}, H∞(Ω) denotes the Banach space of bounded analytic functions in Ω with supremum norm ∥ · ∥H∞ For ζ ∈ Ω, we putwhere f′(∞) = lim,z→∞z{f (∞) = f(z)} if ζ = ∞.

scholarly journals Fourier's series and analytic functions

1922 ◽  
Vol 101 (709) ◽  
pp. 124-133 ◽  
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Real Variable ◽  
The Real ◽  
Real Value

1.1. In the theorems which follow we are concerned with functions f ( x ) real for real x and integrable in the sense of Lebesgue. We do not, however, remain in the field of the real variable, for we suppose, in 4 et seq ., that f ( x ), or a function associated with f ( x ), is analytic, or, at any rate, harmonic, in a region of the complex plane associated with the particular real value of x considered. The Fourier’s series considered are those associated with the interval (0, 2π). If a is a point of the interval, we write ϕ(u) = ½ { f ( a + u ) + f ( a - u ) - 2 s } (0 < a < 2π), (1.11) ϕ(u) = ½ { f ( u ) + f (2π - u ) - 2 s } ( a = 0, a = 2π), (1.12) where s is a constant.

LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS

2016 ◽  
Vol 103 (1) ◽  
pp. 104-115 ◽  
Author(s):  
THOMAS H. MACGREGOR ◽  
MICHAEL P. STERNER
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Power Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Hadamard Product ◽  
Open Unit ◽  
Open Unit Disk

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

scholarly journals The Bloch space and Besov spaces of analytic functions

1996 ◽  
Vol 54 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Elementary Proof ◽  
Bloch Space ◽  
Little Bloch Space ◽  
Spaces Of Analytic Functions

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

scholarly journals ON APPROXIMATION OF ANALYTIC FUNCTIONS BY PERIODIC HURWITZ ZETA-FUNCTIONS

2018 ◽  
Vol 24 (1) ◽  
pp. 20-33 ◽  
Author(s):  
Darius Siaučiūnas ◽  
Violeta Franckevič ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Sequence ◽  
Hurwitz Zeta Function ◽  
Complex Numbers ◽  
Closed Set

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

scholarly journals Analytic capacity for two segments

1991 ◽  
Vol 122 ◽  
pp. 19-42 ◽  
Author(s):  
Takafumi Murai
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Supremum Norm ◽  
Compact Set ◽  
Analytic Capacity ◽  
Bounded Analytic Functions

The analytic capacity γ(E) of a compact set E in the complex plane C is defined by γ(E) = sup , where — f′(∞) is the 1/z-coeffieient of f(ζ) at infinity and the supremum is taken over all bounded analytic functions f(ζ) outside E with supremum norm less than or equal to 1. Analytic capacity γ(·) plays various important roles in the theory of bounded analytic functions.

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