Topic
Inverse trigonometric functions
About: Inverse trigonometric functions is a research topic. Over the lifetime, 854 publications have been published within this topic receiving 11141 citations. The topic is also known as: arcus function & antitrigonometric function.
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TL;DR: In this article, the summation of trigonometric series over integrals involving Bessel or Struve functions is studied, where the series are expressed as power series in terms of Riemann's ζ or Catalan's β function or Dirichlet functions η and λ.
Abstract: To find formulas for the summation of trigonometric series over integrals involving Bessel or Struve functions, we rely on trigonometric series involving Bessel or Struve functions, which are in turn obtained by using summation formulas for series over the product of two trigonometric functions. All these sums are expressed either as power series in terms of Riemann’s ζ or Catalan’s β function or Dirichlet functions η and λ, or, in certain cases, they are brought in so called closed form, which means that the infinite series are represented by finite sums. Important limiting values cases are considered too.
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Abstract: This note presents a simple formula for pi.
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TL;DR: The Ohtsuka-Vălean sum was extended to evaluate an extensive number of trigonometric and hyperbolic sums and products as mentioned in this paper , taking over finite and infinite domains defined in terms of the Hurwitz-Lerch zeta function.
Abstract: The Ohtsuka–Vălean sum is extended to evaluate an extensive number of trigonometric and hyperbolic sums and products. The sums are taken over finite and infinite domains defined in terms of the Hurwitz–Lerch zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products and sums of tangent, cotangent, hyperbolic tangent and hyperbolic cotangent functions, in certain cases raised to a complex number power.
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TL;DR: In this article , the problem of fixed-time tracking control for a class of uncertain Euler-Lagrange systems with modeling uncertainties, actuator faults, and external disturbances is addressed.
Abstract: This article deals with the problem of fixed-time tracking control for a class of uncertain Euler–Lagrange systems with modeling uncertainties, actuator faults, and external disturbances. First, a novel NFTSM variable is developed by introducing an arctan function. This sliding mode can not only eliminate singular values, but also ensure that settling time is irrelevant to the initial conditions of system states. Then, an ANFTSMC law is designed by combining NFTSM and adaptive techniques. The significant features of this approach are that the proposed controller is continuous and chattering free, and the fixed-time stability is ensured in the presence of uncertainties and actuator faults. Theoretical analysis shows that the proposed sliding mode surface and the errors can converge into a small neighborhood in the scheduled time. Finally, the superior of the developed control strategy is substantiated with comparisons of simulation results.