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Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Qi Feng

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

710--736

EN

In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.

2

Mimicking speaker’s lip movement on a 3D head model using cosine function fitting

Lüsi I., Anbarjafari G.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2017

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Vol. 65, nr 5

733--739

EN

Real-time mimicking of human facial movement on a 3D head model is a challenge which has attracted attention of many researchers. In this research work we propose a new method for enhancing the capturing of the shape of lips. We present an automatic lip movement tracking method which employs a cosine function to interpolate between extracted lip features in order to make the detection more accurate. In order to test the proposed method, mimicking lip movements of a speaker on a 3D head model is studied. Microsoft Kinect II is used in order to capture videos and both RGB and depth information are used to locate the mouth of a speaker followed by fitting a cosine function in order to track the changes of the features extracted from the lips.

3

Measure of noncompactness and neutral functional differential equations with state-dependent delay

Benchohra M., Henderson J., Medjadj I.

Journal of Mathematics and Applications

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2016

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Vol. 39

23--45

EN

Our aim in this work is to study the existence of solutions of first and second order for neutral functional differential equations with state-dependent delay. We use the Mönch’s fixed point theorem for the existence of solutions and the concept of measures of noncompactness.

4

On generalized sine and cosine functions, II

Adamaszek-Fidytek I.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

571--578