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Fractional Trigonometry and the Spiral Functions

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Abstract

This paper develops two related fractional trigonometries based on the multi-valued fractional generalization of the exponential function, the R-function. The trigonometries contain the traditional trigonometric functions as proper subsets. Also developed are relationships between the R-function and the new fractional trigonometric functions. Laplace transforms are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Because of the fractional character of the R-function, several new trigonometric functions are required to augment the traditional sine, cosine, etc. functions. Fractional generalizations of the Euler equation are derived. As a result of the fractional trigonometry a new set of phase plane functions, the Spiral functions, that contain the circular functions as a subset, is identified. These Spiral functions display many new symmetries.

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Authors and Affiliations

  1. National Aeronautics and Space Administration, John H. Glenn Research Center, 21000 Brookpark Road, Cleveland, OH, 44135, U.S.A.

    Carl F. Lorenzo

  2. Department of Electrical & Computer Engineering, The University of Akron, Akron, OH, 44325-3904, U.S.A.

    Tom T. Hartley

Authors
  1. Carl F. Lorenzo
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  2. Tom T. Hartley
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Correspondence to Carl F. Lorenzo.

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Lorenzo, C., Hartley, T. Fractional Trigonometry and the Spiral Functions. Nonlinear Dyn 38, 23–60 (2004). https://doi.org/10.1007/s11071-004-3745-9

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  • DOI: https://doi.org/10.1007/s11071-004-3745-9

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