“Generalized Hyperbolic Functions”的概念、定义、习题解题方法及翻译-数学辞典

单词

Generalized Hyperbolic Functions

释义

Generalized Hyperbolic Functions

In 1757, V. Riccati first recorded the generalizations of the Hyperbolic Functions defined by

(1)

for

, ...,

, where

is Complex, with the value at

defined by

(2)

This is called the

-hyperbolic function of order

of the

th kind. The functions

satisfy

(3)

where

(4)

In addition,

(5)

The functions give a generalized Euler Formula

(6)

Since there are

th roots of

, this gives a system of

linear equations. Solving for

gives

(7)

where

(8)

is a Primitive Root of Unity.

The Laplace Transform is

(9)

The generalized hyperbolic function is also related to the Mittag-Leffler Function

(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

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.

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