Integration Of Analytic Functions Research Articles

Stationary measures associated to analytic iterated function schemes

AbstractWe study how the stationary measure associated to analytic contractions on the unit interval behaves under changes in the contractions and the weights. Firstly we give a simple proof of the fact that the integrals of analytic functions with respect to the stationary measure vary analytically if we perturb the contractions and the weights analytically. Secondly, we consider the special case of affine contractions and we prove a conjecture of J. Fraser in on the Kantorovich–Wasserstein distance between two stationary measures associated to affine contractions on the unit interval with different rates of contraction.

Approximation of analytic functions in adaptive environment

In this paper, a mixed quadrature rule of degree of precision seven is formed for analytic functions by taking two constituent rules each of degree of precision five. Here the integral of analytic function is converted to real definite integrals with the help of double transformations. Then the mixed quadrature rule is tested in adaptive environment and it is obviously superior to that of Gauss–Legendre three-point rule.

AN EXTRAPOLATORY QUADRATURE RULEFOR ANALYTIC FUNCTIONS

A quadrature rule for the numerical evaluation of integrals of analytic functions along directed line segments in the complex plane has been formulated using the transformed rule based on Gauss Legendre two point quadrature formulas and an interpolatory three point rule. The degree of precision has been increased from five to seven

Numerical computation of integral of analytic functions in complex plane

A family of interpolatory type of quadrature rules of degree of precision at least seven has been constructed for the approximate evaluation of contour integrals of analytic functions along a directed line segment in the complex plane. The relative accuracies of rules have been studied in their respective classes and also numerically verified by integrating some standard test integrals.

Numerical computation of integral of analytic functions in complex plane

Numerical computation of integral of analytic functions in complex plane

A Mixed quadrature rule for numerical integration of analytic functions by using fejer and gaussian quadrature rules

In this paper 5-point Fejer’s second rule has been modified for computing integrals involving analytic functions on a line. This rule has been mixed with Gaussian 3-point rule to form a mixed quadrature rule. The relative efficiencies of this rule with respect to its constituent rules and other previously developed mixed rules have been numerically verified on test integrals. Asymptotic error estimate of this rule has been determined.

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n = m = 2 we determine the maximum number of critical periods and their location; to the case n = m = 3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.

On the Evaluation of Correction Terms of Gaussian Integration

Accurate and compact computing recipe for evaluating correction terms for the Gaussian integration of analytic functions offers economy and high accuracy for many integrals of interest in physical sciences. The requirement of numerical integration in physical sciences is quite pervasive because many practical functions aren't integrable in closed form. A simple example is the error function Erβx) defined by1 Erf(x) 2 /<; dt.

Elastic complex analysis and its applications in fracture mechanics

The key point in this paper is the introduction of elastic analytic functions. An elastic analytic function is a function of the form u : C → C 2 which is differentiable and satisfies equations which are analogous to the Cauchy–Riemann equations of traditional complex analysis such that the following conditions hold: first the real and imaginary part of the first complex component of u satisfy the Navier equation of plane elasticity, and second, the derivative along a line of the real and imaginary part of the second complex component of u is proportional to the applied tractions along that line. Algebraical operations have been defined on elastic analytic functions such that they constitute a commutative algebra over the real field and a module over the set of analytic functions. Next, a derivative and an integral of elastic analytic functions are introduced such that they behave in a similar way to complex differentiation and integration of analytic functions, in particular we have properties such as: the integral of an elastic analytic function around a contour is zero, a Cauchy-like integral formula and Plemelj-like formulae. These properties can be very useful in tackling problems of plane elasticity involving cracks through the boundary element method. It is also proved that path independent integrals in plane elasticity that are derived from Noether’s theorem, whose integrand only depends on the position and gradient of displacements, can be written as the integral of an elastic analytic function.

Formulas for the numerical approximation of contour integrals

Some quadrature formulas of degree of precision nine, eleven and thirteen for the numerical evaluation of contour integrals of analytic functions have been constructed.

Numerical quadrature of analytic functions

A four‐point quadrature rule with degree of p recision five for the numerical approximation of integrals of analytic functions of a complex variable has been constructed. An asymptotic error estimate and an error bound of the rule constructed have been derived.

Numerical evaluation of integrals of analytic functions

Numerical evaluation of integrals of analytic functions

Automatic, guaranteed integration of analytic functions

Davis introduced a method for estimating linear functionals of analytic functions by using Cauchy's Integral Formula. This is used to construct methods for numerical integration which give rigorous error bounds. By combining these bounds with strategies for order and subinterval adaptation, a program is developed for automatic integration of analytic functions. Interval analysis is used to validate the bounds.

Approximation of double integrals of analytic functions of two complex variables

A nine-point degree five rule for the numerical approximation of double integrals of analytic functions of complex variables and its error bound have been derived.

Approximate evaluation of multiple complex integrals of analytic functions

For the numerical evaluation of multiple integrals of analytic functions ofn complex variables, a (4n+1)-point degree five non-product rule has been constructed discarding certain points from the set of 5n nodes needed in the product layout based on the five point degree five rule due to Birkhoff and Young. An asymptotic error estimate for the casen=2 has been determined.

Compound Birkhoff‐Young rule for numerical integration of analytic functions

A compound rule for the numerical integration of an analytic function along a directed line segment in the complex plane, basing on the Birkhoff‐Young 5‐point formula has been formulated. The rule has been tested in case of some examples and the convergence of the computed approximate values to the exact values of the integrals has been asserted.