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On fractional derivatives of trigonometric functions and exponential functions.
Chaos: An Interdisciplinary Journal of Nonlinear ScienceIn some places, fractional derivative is abbreviated as Dα, dαdtα(ordαdxα), where α is positive but not an integer. The “formulas” dαsintdtα=sin(t+πα2), dαcostdtα=cos(t+πα2), and dαeλtdtα=λαeλt are sometimes used. However, they generally hold for α∈Z+={0,1,2,…}.
Changpin Li, Jianhua Tang
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Exponential and Trigonometric Functions—From the Book
The Mathematical Intelligencer, 2003Martin D. Davis
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Integral Transforms and Special Functions, 2008
Abstract In this paper, we obtain two Gaussian hypergeometric summation formulas and show how each of these summation formulas can be applied to derive several general double-series identities and various generalized hypergeometric representations of such combinations of the exponential and trigonometric functions as The results presented here are ...
M. I. Qureshi +2 more
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Abstract In this paper, we obtain two Gaussian hypergeometric summation formulas and show how each of these summation formulas can be applied to derive several general double-series identities and various generalized hypergeometric representations of such combinations of the exponential and trigonometric functions as The results presented here are ...
M. I. Qureshi +2 more
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Bounding the gamma function in terms of the trigonometric and exponential functions
Acta Scientiarum Mathematicarum, 2017Feng Qi (祁锋), Mansour Mahmoud
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The generalized exponential function and fractional trigonometric identities
2011 20th European Conference on Circuit Theory and Design (ECCTD), 2011In this work, we recall the generalized exponential function in the fractional-order domain which enables defining generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize them from the narrow integer-order subset to the more general fractional-order domain. Generalized hyperbolic function relations
A. G. Radwan, A. S. Elwakil
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On an exponential-trigonometric natural interpolation spline
IEEE/OES Working Conference on Current Measurement Technology, 2021In the present paper, using the discrete analogue of the operator d8/dx8 + 2d4/dx4 + 1, an interpolation spline that minimizes the quantity ∫01(φIV(x)+φ(x))2dx in the Hilbert space W2(4,0) is constructed.
A. Boltaev, D. Akhmedov
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A Scalable Frequency Estimation Method Based on Multi-Point Interpolation of Trigonometric Functions
IEICE Transactions on Fundamentals of Electronics Communications and Computer SciencesSUMMARY Interpolation-based frequency estimation methods can be used to improve the frequency estimation accuracy of discrete Fourier transform (DFT) methods for complex exponential or real sinusoidal signals.
Li Cheng, Huaixing Wang
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Defining Exponential and Trigonometric Functions Using Differential Equations
Mathematics Magazine, 2014SummaryThis note addresses the question of how to rigorously define the functions exp(x), sin(x), and cos(x), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem.
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A NOTE ON TWO RESULTS ASSOCIATED WITH EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
Far East Journal of Mathematical Sciences (FJMS), 2016Summary: The objective of this note is to show how we can establish Taylor-Maclaurin series expansions of products of exponential and trigonometric functions in an elementary way.
Sukanya, M. +2 more
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Real Exponential, Logarithmic, and Trigonometric Functions for Physicists
2022Real Exponential, Logarithmic, and Trigonometric Functions for Physicists builds on the author's previous books focused on mathematics for scientists. This new work presents a rigorous study of the underlying mathematical functions used in physics and provides its readers a deeper understanding for those studying and practicing the physical sciences.
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