In 1757, V. Riccati first recorded the generalizations of the Hyperbolic Functions defined by
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
The Laplace Transform is
(9) |
(10) |
The values and give the exponential and circular/hyperbolic functions (depending on the sign of ),
respectively.
(11) | |||
(12) | |||
(13) |
See also Hyperbolic Functions, Mittag-Leffler Function
References
Kaufman, H. ``A Biographical Note on the Higher Sine Functions.'' Scripta Math. 28, 29-36, 1967.
Muldoon, M. E. and Ungar, A. A. ``Beyond Sin and Cos.'' Math. Mag. 69, 3-14, 1996.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996.
Ungar, A. ``Generalized Hyperbolic Functions.'' Amer. Math. Monthly 89, 688-691, 1982.
Ungar, A. ``Higher Order Alpha-Hyperbolic Functions.'' Indian J. Pure. Appl. Math. 15, 301-304, 1984.
© 1996-9 Eric W. Weisstein