Inverse Sine Function Research Articles

Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

Prescribed-Time Fractional Order Control of DWR via Time-Varying Scaling Function

This study focuses on a differential wheeled robot’s (DWR) prescribed-time fractional order position control. Firstly, based on the kinematic model of DWR, a distance-related orientation error is designed using the inverse sine function. Based on this, an improved linear velocity constraint function is proposed. Compared with existing methods, while ensuring the correlation between velocity and orientation error, the multibalance point risk caused by large angle errors is avoided. Then, a prescribed-time fractional order position controller based on a time-varying scaling function is proposed to stabilize the kinematic system of DWR in the prescribed time. This controller can stabilize the position control system of the DWR in a prescribed time by adjusting the prescribed-time parameter, avoiding the infinite gain risk in traditional prescribed-time controllers. Finally, through numerical simulation, we verify that the proposed control law can converge the system status of DWR to the bounded interval in the prescribed time.

A mathematical method to estimate angle and distance for percutaneous renal puncture based on computed tomography data: Description and validation.

Gaining access to the kidney is crucial step in percutaneous nephrolithotomy (PCNL); it has a steep learning curve. Describe the mathematical method to predict renal puncture angle and distance based on preoperative computed tomography (CT) measurements. Then evaluating how it correlates with measured values. The study was prospectively designed. After ethical committee approval, the study uses data from preoperative CT to construct a triangle so we can estimate puncture depth and angle. A triangle of three points, the first is point of entry to the pelvicalyceal system (PCS), the second is point on the skin perpendicular to it, and the third where the needle punctures the skin. The needle travel is estimated using the Pythagorean theorem and puncture angle using the inverse sine function. We evaluated 40 punctures in 36 PCNL procedures. After PCS puncture using fluoroscopy-guided triangulation, we measured the needle travel distance and angle to the horizontal plane. Then compared the results with mathematically estimated values. We targeted posterior lower calyx in 21 (70%) case. The correlation between measured and estimated needle travel distance with Rho coefficient of 0.76 with P < 0.001. The mean difference between the estimated and the measured needle travel was - 0.37 ± 1.2 cm (-2.6-1.6). Measured and estimated angle correlate with Rho coefficient of 0.77 and P < 0.001. The mean difference between the estimated and the measured angle was 2° ± 8° (-21°-16°). Mathematical estimation of needle depth and angle for gaining access to the kidney correlates well with measured values.

Several Double Inequalities for Integer Powers of the Sinc and Sinhc Functions with Applications to the Neuman–Sándor Mean and the First Seiffert Mean

In the paper, the authors establish a general inequality for the hyperbolic functions, extend the newly-established inequality to trigonometric functions, obtain some new inequalities involving the inverse sine and inverse hyperbolic sine functions, and apply these inequalities to the Neuman–Sándor mean and the first Seiffert mean.

The Natural Approaches of Shafer-Fink Inequality for Inverse Sine Function

In this paper, we obtain some new natural approaches of Shafer-Fink inequality for arc sine function and the square of arc sine function by using the power series expansions of certain functions, which generalize and strengthen those in the existing literature.

MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

Harmonic-binomial Euler-like sums via expansions of $$(\arcsin x)^p$$

In this paper we evaluate Euler-like sums involving harmonic/odd harmonic numbers and central binomial coefficients by using the expansion for powers of the inverse sine function. We also evaluate a number of integrals which in turn give rise to new central binomial coefficient and harmonic number identities.

A necessary and sufficient condition for the convexity of the one-parameter generalized inverse trigonometric sine function according to power mean

A necessary and sufficient condition for the convexity of the one-parameter generalized inverse trigonometric sine function according to power mean

Inequalities related to certain inverse trigonometric and inverse hyperbolic functions

In this paper, we obtain a sharp double inequality between the inverse tangent and inverse hyperbolic sine functions. At the same time, we give a sharp double inequality between the inverse hyperbolic tangent and inverse sine functions.

Some new upper and lower bounds for the Mills ratio

In this paper, we present new upper and lower bounds for the Mills ratio of the standard Gaussian law. Several different methods are used to derive these new bounds. One of the methods reproduces the bounds of several different authors in previous works as special cases and is a very general method that produces many new bounds. One of the bounds can be written in terms of hyperbolic sine and inverse hyperbolic sine functions. Some of the bounds involve exponential functions and are improved versions of previously proposed bounds or are improved versions of the new bounds introduced earlier in this paper. Some results from reliability theory and Jensen's inequality are used to improve determinantal inequalities. Some open problems are discussed, and conjectures are made.

Sharp one-parameter geometric and quadratic means bounds for the Sándor–Yang means

In the article, we present the best possible upper and lower bounds for the Sandor–Yang means in terms of the families of one-parameter geometric and quadratic means, and discover new bounds for the inverse tangent and inverse hyperbolic sine functions.

A digital-mixing-based method for timing skew estimation in time-interleaved ADCs

Time-interleaving has been a popular choice for multi-GHz analog-to-digital converters (ADCs). Unfortunately, inherent defects such as offset, gain, timing-skew mismatches among sub-ADCs degrade overall performance seriously. At present, the method for eliminating offset and gain mismatch is fairly straightforward, however, calibration for timing-skew is still in a state of exploration. An efficient digital-mixing-based method to estimate timing skew in time-interleaved analog-to-digital converters (TI-ADCs) is proposed and verified. The timing skew can be estimated by digital-mixing and inverse sine function processing. The proposed method can greatly alleviate the effects of phase ambiguity. It is also insensitive to offset mismatch. A four-channel TI-ADC behavioral system of 10-bits 2 GS/s has been simulated to verify the performance of the method. Simulation results demonstrate that the method greatly improves the TI-ADC′s performance and is effective up to the severe timing skew mismatch.

Dual Taylor Series, Spline Based Function and Integral Approximation and Applications

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series approximation. It is shown that a spline based series approximation to an integral yields, in general, a higher accuracy for a set order of approximation than a dual Taylor series, a Taylor series and an antiderivative series. A spline based series for an integral has many applications and indicative examples are detailed. These include a series for the exponential function, which coincides with a Padé series, new series for the logarithm function as well as new series for integral defined functions such as the Fresnel Sine integral function. It is shown that these series are more accurate and have larger regions of convergence than corresponding Taylor series. The spline based series for an integral can be used to define algorithms for highly accurate approximations for the logarithm function, the exponential function, rational numbers to a fractional power and the inverse sine, inverse cosine and inverse tangent functions. These algorithms are used to establish highly accurate approximations for π and Catalan’s constant. The use of sub-intervals allows the region of convergence for an integral approximation to be extended.

High-accuracy resolver-to-linear signal converter

ABSTRACTA high-accuracy resolver-to-linear signal converter for the measurement of angular displacement is proposed in this paper. The proposed converter comprises two sections: a demodulator and linear shaper. In the first section, the demodulator makes use of the sample-and-hold circuit (SHC) to sample the peak amplitude of the resolver signal. The control signal of the SHC is provided from the resolver signals instead of the excitation signal used in traditional approaches. The proposed demodulator requires no analogue multiplier and low-pass filter. Therefore, the fast response time of the proposed demodulator is achieved. In the second section, the linear shaper consists of the inverse-sine function scheme together with a switched-gain amplifier to produce the linear signal proportional to the shaft angle. The hyperbolic tangent characteristic of the operational transconductance amplifier is utilised to realise the inverse-sine function scheme. The proposed technique requires one phase of the resolver signal to obtain the linear signal. Therefore, the position error caused by amplitude imbalance between the two resolver signals is avoided. The performances of the proposed converter are discussed in detail and demonstrated by an experimental implementation using commercial devices. The experimental results show that the maximum relative error and response time for the excitation frequency of 3 kHz are measured as 0.06% and 0.11 ms, respectively.

The Ambiguity of the Inverse Secant

Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.

Refinements and generalizations of some inequalities of Shafer-Fink\u2019s type for the inverse sine function

In this paper, we give some sharper refinements and generalizations of inequalities related to Shafer-Fink’s inequality for the inverse sine function stated in Theorems 1, 2, and 3 of Bercu (Math. Probl. Eng. 2017: Article ID 9237932, 2017).

Sharp Refinements for the Inverse Sine Function Related to Shafer-Fink’s Inequality

The aim of our work is to give new sharp refinements of Shafer-Fink’s inequality, using suitable changes of variables.

The inverse hyperbolic tangent function and Jacobian sine function

The Hölder convexity of the composition of inverse hyperbolic tangent function and Jacobian sine function is investigated in this paper.

Sharp Cusa type inequalities with two parameters and their applications

In the article, we present several sharp Cusa type inequalities with two parameters. As applications, some new Shafer–Fink type and Carlson type inequalities for inverse sine and cosine functions, and new inequalities for bivariate means are found.

On the Inverse Sine Function, π and Babylonian Identity

We evaluate the constant π using the Babylonian identity and the inverse sine function.