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Analysis of inductance calculation of coaxial circular filamentary coils, thin-wall solenoids, and disk coils using inverse hyperbolic functions

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IET SCIENCE MEASUREMENT & TECHNOLOGY 2016年第7期10卷 754-760页

作者： Liang, Sizhuang Zhejiang Univ Coll Elect Engn Hangzhou Zhejiang Peoples R China

This study presents a method using inverse hyperbolic functions to calculate the mutual and self-inductance inductance of coaxial circular filamentary coils, thin-wall solenoids, disk coils, and coils with rectangular cross-section in air. A generalised expression is given for any combination of these coils, which only requires numerical integration with one variable. Singularities are fully analysed and methods to resolve the singularities are presented in this study. When calculating the self-inductance of a disk coil or a thin-wall solenoid, the proposed method involves an improper integral, and the accuracy of the proposed method is limited unless a quadrature rule applicable to improper integrals is applied. For other situations, results by the proposed method were compared with those in the literature, and they were in excellent agreement.

关键词： hyperbolic equations solenoids inductance inverse hyperbolic functions mutual inductance self-inductance coaxial circular filamentary coils thin-wall solenoids disk coils rectangular cross-section

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Analysis of Inductance Calculation of Coaxial Circular Coils With Rectangular Cross Section Using inverse hyperbolic functions

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY 2015年第4期25卷 1-9页

作者： Liang, Sizhuang Fang, Youtong Zhejiang Univ Coll Elect Engn Hangzhou 310027 Zhejiang Peoples R China

This paper presents an improved method using inverse hyperbolic functions to calculate the mutual inductance and self-inductance of coaxial circular coils with rectangular cross section and uniform current density in air. The method is simple, fast, and easy to use, which only entails numerical integration with one variable. The method is derived by using inverse hyperbolic functions but can be extended to using logarithmic functions. An algorithm and a leading part checking method to resolve singularities are presented, and their theoretical foundations are fully analyzed. The results of several testing cases calculated by the proposed method were compared with those in the literature, and they were in excellent agreement.

关键词： Circular coils inductance inverse hyperbolic functions numerical stability

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MATRIX inverse TRIGONOMETRIC AND inverse hyperbolic functions: THEORY AND ALGORITHMS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2016年第4期37卷 1453-1477页

作者： Aprahamian, Mary Higham, Nicholas J. Univ Manchester Sch Math Manchester M13 9PL Lancs England

Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and the principal values acos, asin, acosh, and asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a "round trip" formula that relates acos(cosA) to A and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pade approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas.

关键词： matrix function inverse trigonometric functions inverse hyperbolic functions matrix inverse sine matrix inverse cosine matrix inverse hyperbolic sine matrix inverse hyperbolic cosine matrix exponential matrix logarithm matrix sign function rational approximation Pade approximation MATLAB GNU Octave Frechet derivative condition number

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Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions

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INTEGRAL TRANSFORMS AND SPECIAL functions 2012年第12期23卷 865-873页

作者： Chen, Chao-Ping Henan Polytech Univ Sch Math & Informat Jiaozuo City 454003 Henan Province Peoples R China

We establish sharp Wilker and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions.

关键词： inequalities trigonometric functions hyperbolic functions inverse trigonometric functions inverse hyperbolic functions

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Convolution identities of poly-Cauchy numbers with level 2

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RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA 2022年 148卷 245-262页

作者： Komatsu, Takao Zhejiang Sci Tech Univ Dept Math Sci Sch Sci Hangzhou 310018 Peoples R China

- Poly-Cauchy numbers with level 2 are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we introduce the Stirling numbers of the first kind with level 2 in order to establish some relations with poly -Cauchy numbers with level 2. Then, we show several convolution identities of poly-Cauchy numbers with level 2. In particular, that of three poly-Cauchy numbers with level 2 can be expressed as a simple form.

关键词： - Poly-Cauchy numbers hyperbolic functions inverse hyperbolic functions convolutions Stirling numbers of the first kind

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Wilker- and Huygens-type inequalities and solution to Oppenheim's problem

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INTEGRAL TRANSFORMS AND SPECIAL functions 2012年第5期23卷 325-336页

作者： Chen, Chao-Ping Cheung, Wing-Sum Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China Henan Polytech Univ Sch Math & Informat Jiaozuo City 454003 Henan Province Peoples R China

We establish Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions. We also provide a laconic proof to Oppenheim's problem associated with inequalities involving the sine... 详细信息

关键词： Wilker inequalities Huygens inequalities inverse trigonometric functions inverse hyperbolic functions

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SOME MISCELLANEOUS PROPERTIES AND APPLICATIONS OF CERTAIN OPERATORS OF FRACTIONAL CALCULUS

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TAIWANESE JOURNAL OF MATHEMATICS 2010年第6期14卷 2469-2495页

作者： Lin, Shy-Der Srivastava, H. M. Univ Victoria Dept Math & Stat Victoria BC V8W 3R4 Canada Chung Yuan Christian Univ Dept Appl Math Chungli 32023 Taiwan Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields. The main object of this paper is to consider some miscellaneous properties and applications which are associated with several fractional differintegral operators. We first investigate, in a systematic and unified manner, various families of series identities which emerged in connection with some of these fractional differintegral formulas. By using such operators of fractional calculus, a number of integral formulas as well as fractional differintegral formulas involving inverse hyperbolic functions are also evaluated.

关键词： Fractional calculus Fractional differintegral operators Series identities Integral formulas inverse hyperbolic functions Gamma function Power functions Composite functions Rational functions Fractional differintegral formulas Generalized hypergeometric functions Fox-Wright functions Gauss hypergeometric function nth derivative formula Generalized Leibniz rule Analytic functions Index law Linearity property Principal value F and H functions Hypergeometric reduction formulas Legendre's duplication formula Incomplete Gamma function Harmonic numbers Exponential integrals Combinatorial identity Psi (or Digamma) function

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Poly-Cauchy numbers with level 2

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INTEGRAL TRANSFORMS AND SPECIAL functions 2020年第7期31卷 570-585页

作者： Komatsu, Takao Pita-Ruiz, Claudio Zhejiang Sci Tech Univ Sch Sci Dept Math Sci Hangzhou 310018 Zhejiang Peoples R China Univ Panamer Fac Ingn Ciudad De Mexico Mexico

We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and logarithms. On the contrary, poly-Bernoulli numbers can be from the inverse relation of logarithms and exponentials. In this similar stream, poly-Cauchy numbers with level 2 may be yielded from the inverse relation about the hyperbolic sine function, which is a 2-step function of the exponential function. In this article, we show several expressions, relations, and properties about poly-Cauchy numbers with level 2. Poly-Cauchy numbers with level 2 can be expressed in terms of multinomial coefficients with combinatorial summation, Stirling numbers of the first kind, or iterated integrals. We also give some recurrence relations for poly-Cauchy numbers with level 2. When the index is negative, the double summation may be formulated as a closed form. A simple case of Cauchy numbers with level 2 has some more relations with D numbers from higher-order Bernoulli numbers or complementary Euler numbers. We prove some more expressions in determinants, continued fractions or by Trudi's formula.

关键词： Poly-Cauchy numbers recurrences hyperbolic functions inverse hyperbolic functions determinants continued fractions

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An algorithm for computing logarithms and arctangents

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Mathematics of Computation 1972年第118期26卷 543-549页

作者： Carlson, B.C. Departments of Physics and Mathematics Iowa State University United States

An iterative algorithm with fast convergence can be used to compute logarithms, inverse circular functions, or inverse hyperbolic functions according to the choice of initialconditions. Only rational operations and square roots are required. The method consists in adding an auxiliary recurrence relation to Borchardt's algorithm to speed the convergence. © 1972 American Mathematical Society.

关键词： Acceleration of convergence Arctangents Borchardt's algorithm inverse circular functions inverse hyperbolic functions Iteration Logarithms

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