Generalized Hyperbolic Functions
المؤلف:
Kaufman, H.
المصدر:
"A Biographical Note on the Higher Sine Functions." Scripta Math. 28
الجزء والصفحة:
...
3-6-2019
1884
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
![f^((k))(0)=<span style=]() {0 k!=r, 0<=k<=n-1,; 1 k=r. " src="http://mathworld.wolfram.com/images/equations/GeneralizedHyperbolicFunctions/NumberedEquation4.gif" style="height:41px; width:198px" /> |
(4)
|
In addition,
![d/(dx)F_(n,r)^alpha(x)=<span style=]() {F_(n,r-1)^alpha(x) for 0<r<=n-1; alphaF_(n,n-1)^alpha(x) for r=0. " src="http://mathworld.wolfram.com/images/equations/GeneralizedHyperbolicFunctions/NumberedEquation5.gif" style="height:46px; width:261px" /> |
(5)
|
The functions give a generalized Euler formula
![e^(RadicalBox[alpha, n])=sum_(r=0)^(n-1)(RadicalBox[alpha, n])^rF_(n,r)^alpha(x).](http://mathworld.wolfram.com/images/equations/GeneralizedHyperbolicFunctions/NumberedEquation6.gif) |
(6)
|
Since there are 
th roots of 
, this gives a system of 
linear equations. Solving for 
gives
![F_(n,r)^alpha(x)=1/n(RadicalBox[alpha, n])^(-r)sum_(k=0)^(n-1)omega_n^(-rk)exp(omega_n^kRadicalBox[alpha, n]x),](http://mathworld.wolfram.com/images/equations/GeneralizedHyperbolicFunctions/NumberedEquation7.gif) |
(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 
by
The values 
and 
give the exponential and circular/hyperbolic functions (depending on the sign of 
), respectively.
In particular
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.
Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.
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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