Continued Fraction Approximation Research Articles

Convergence Acceleration of Infinite Series Involving the Product of Riccati–Bessel Function and Its Application for the Electromagnetic Field: Using the Continued Fraction Expansion Method

A summation technique has been developed based on the continuous fractional expansion to accelerate the convergence of infinite series involving the product of Riccati–Bessel functions, which are common to electromagnetic applications. The series is transformed into a new and faster convergent sequence with a continued fraction form, and then the continued fraction approximation is used to accelerate the calculation. The well-known addition theorem formula for spherical wave function is used to verify the correctness of the algorithm. Then, some fundamental aspects of the practical application of continuous fractional expansion for Mie scattering theory and electromagnetic exploration are considered. The results of different models show that this new technique can be applied reliably, especially in the electromagnetic field excited by the vertical electric dipole (VED) source in the “earth-ionospheric” cavity. The comparison among the new technology, the Watson-transform, and the spherical harmonic series summation algorithm shows that this new technology only needs less than 120 series items which is already enough to obtain a small relative error, which greatly improves the convergence speed, and provides a new way to solve the problem.

A method to construct continued-fraction approximations and its applications

In this paper, we provide a method to construct a continued-fraction approximation based upon a given asymptotic expansion. As applications of the method developed here, we establish several continued-fraction approximations for the gamma and the digamma (or psi) functions.

Some new properties of the Barnes G-function and related results

In this paper, we present several potentially useful properties of the Barnes G-function. The properties considered here include, for example, its integral representation, complete monotonicity, and continued-fraction approximation. We also derive continued-fraction approximations of the Glaisher-Kinkelin constant and the Choi-Srivastava constants.

Adaptive fractional‐order control of power system frequency in the presence of wind turbine

Inertial and droop control are common methods for improving grid frequency control. Using the first-order derivative of the frequency in the inertial control loop only investigates the rate of frequency variation, while other useful information is not taken into consideration. Accordingly, in this study, the application of a fractional-order derivative (between 1 and 2) in the inertial control loop is proposed to have more useful frequency-based information. To achieve this goal, firstly continued fraction approximations and secondly, the discretization is adopted by using the Tustin Method with Prewarping. Inertia loop and droop loop gain have a significant impact on the operation of the wind turbines and the power system frequency control. Therefore, in this paper, a novel method for the adaptive adjusting of droop and inertia control loops gains in DFIG is proposed. To utilize more frequency error information, fractional-order adaptive law is proposed to update the loop gains and the inertia control loop. In order to have a robust system, which is permanently stable for every fractional order (FO), is used parameter space method. Simulation results demonstrate the proper performance of the FO-based adaptive approach to increase the frequency nadir (FN) and decrease the frequency variations.

Asymptotic expansions and continued fraction approximations for harmonic numbers

In this paper, we provide a method for constructing a continued fraction approximation based on a given asymptotic expansion. We establish some asymptotic expansions for the harmonic numbers which employ the nth triangular number. Based on these expansions, we derive the corresponding continued fraction approximations for the harmonic numbers.

Almost Mathieu operators with completely resonant phases

Let $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$, where $p_{n}/q_{n}$ is the continued fraction approximation to $\unicode[STIX]{x1D6FC}$. Let $(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $\ell ^{2}(\mathbb{Z})$, where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$. Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170(1) (2009), 303–342] conjectured that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$, $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$. In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$, $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$.

Continued Fractions and Conformal Mappings for Domains with Angel Points

Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.

Approximation formulas related to Somos\u2019 quadratic recurrence constant

We present two classes of asymptotic expansions related to Somos’ quadratic recurrence constant and provide the recursive relations for determining the coefficients of each class of the asymptotic expansions by using Bell polynomials and other techniques. We also present continued fraction approximations related to Somos’ quadratic recurrence constant.

On Some Convergence to the Constant $$\varvec{e}$$ e and Proof of Keller’s Limit

A new asymptotic formula for the constant e is proposed by the continued fraction approximation. As a consequence, we deduce inequalities for the sequence \((1+1/n)^n\), and show an application of the inequalities to the proof of Keller’s limit. Numerical computations and comparisons are conducted to illustrate the superiority of our new inequalities.

Continued Fraction Approximation and Inequality of the Gamma Function

In this paper, we present a continued fraction approximation for the gamma function. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.

Sufficient Conditions for the Equivalent Convergence of Sequences of Different Approximants for Two-Dimensional Continued Fractions

By the method of fundamental inequalities, we study the equivalent convergence of two figured approximants for two-dimensional continued fractions, namely, the approximants obtained from the problem of correspondence of two-dimensional continued fractions to some formal double power series and the problem of equivalence of two-dimensional continued fractions. We also establish sufficient conditions under which the indicated two approximants of two-dimensional continued fractions converge to the same limit.

Continued Fraction Approximation for the Gamma Function by the Tri-gamma Function

In this paper, we present a continued fraction product approximation for the Gamma function, via the Tri-gamma function. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.

Continued Fractions and Inequalities About $$\left( 1+\frac{1}{x}\right) ^x$$ 1 + 1 x x

In this paper, based on the prior works, some general continued fraction approximations and inequalities about \(\left( 1+\frac{1}{x}\right) ^x\) are established. Finally, to demonstrate the superiority of our new series, some numerical computations are given.

Continued fraction approximation for the Gamma function based on the Tri-gamma function

In this paper, we present a continued fraction product approximation for the Gamma function, via the Tri-gamma function. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.

Sharp approximation formulas and inequalities for the Wallis ratio by continued fraction

In this paper, we present a new continued fraction approximation of the Wallis ratio. This approximation is fast in comparison with the recently discovered asymptotic series. We also establish the inequalities related to this approximation. Finally, some numerical computations are provided for demonstrating the superiority of our approximation.

A Kind of New Continued Fraction Approximation of Gamma Function Based on Mortici’s Formula

In this paper, based on Mortici’s works, we establish two sharp general continued fraction approximations for the gamma function. Then related inequalities are given. Finally, for demonstrating the superiority of our new estimations and inequalities, we provide some numerical comparisons.

A new continued fraction approximation and inequalities for the Gamma function via the Tri-gamma function

The main objective of this paper is to propose a product approximation for the Gamma function in the form of continued fraction, via the Tri-gamma function. The importance of this new formula is that the convergence of the corresponding asymptotic series is faster than other recently discovered asymptotic series, and some new inequalities related to the new formula are also established.

Some quicker continued fraction approximations and inequalities towards Euler's constant

In this paper, we define some new continued fraction sequences towards Euler's constant and two related inequalities. We also present some numerical simulations to demonstrate the superiority of the optimal new sequences over new sequences with other coefficients and Lu's sequences at the end of this article.

A nonlinear second derivative method with a variable step-size based on continued fractions for singular initial value problems

We propose a nonlinear one-step second derivative method (NSDM) with a variable step-size implementation based on continued fractions for the numerical solution of singular initial value problems (IVPs). The singular IVPs typically originate from models in Mathematical Biology which are represented by reaction diffusion partial differential equations (PDEs). These PDEs are then discretized in space through finite difference methods to yield a system of ordinary differential equations with singular behavior associated with finite time blow-up. We are motivated by the fact that numerical methods based on polynomial approximation perform poorly in the proximity of a singular point when applied to singular IVPs. Therefore, a continuous method based on a finite continued fraction approximation is derived and used to obtain the NSDM. The NSDM is implemented in a self-starting mode and in a predictor-corrector mode, equipped with a variable step-size implementation. The superiority of the NSDM over standard meth...

Continued fraction approximations and inequalities for the gamma function by Burnside

In this paper, based on the Burnside’s formula, some accurate approximation and double inequality for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside’s formula and other classical or recently discovered asymptotic series, some numerical computations are provided.