New definitions of exponential, hyperbolic and trigonometric functions on time scales
Copyright policy )New definitions of exponential, hyperbolic and trigonometric functions on time scales
We propose two new definitions of the exponential function on time scales. The first definition is based on the Cayley transformation while the second one is a natural extension of exact discretizations. Our eponential functions map the imaginary axis into the unit circle. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way. The resulting functions preserve most of the qualitative properties of the corresponding continuous functions. In particular, Pythagorean trigonometric identities hold exactly on any time scale. Dynamic equations satisfied by Cayley-motivated functions have a natural similarity to the corresponding diferential equations. The exact discretization is less convenient as far as dynamic equations and differentiation is concerned.
27 pages
measure chains, first and second order dynamic equations, Real analysis on time scales or measure chains, time scales, Applied Mathematics, Exponential function, Cayley transform, First and second order dynamic equations, exponential function, trigonometric functions, hyperbolic functions, Time scales, 33B10, 26E70, 34N05, 65L12, Hyperbolic functions, Dynamic equations on time scales or measure chains, exact discretization, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Measure chains, Exact discretization, Analysis, Trigonometric functions
measure chains, first and second order dynamic equations, Real analysis on time scales or measure chains, time scales, Applied Mathematics, Exponential function, Cayley transform, First and second order dynamic equations, exponential function, trigonometric functions, hyperbolic functions, Time scales, 33B10, 26E70, 34N05, 65L12, Hyperbolic functions, Dynamic equations on time scales or measure chains, exact discretization, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Measure chains, Exact discretization, Analysis, Trigonometric functions
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