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) |

For , the first few functions are

**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

1999-05-25