OpenMath Content Dictionary: gen_hyperbolic1

Canonical URL:
http://www.openmath.org/cd/gen_hyperbolic1.ocd
CD File:
gen_hyperbolic1.ocd
CD as XML Encoded OpenMath:
gen_hyperbolic1.omcd
Defines:
generalised_hyperbolic
Date:
2002-11-11
Version:
0 (Revision 1)
Review Date:
2017-12-31
Status:
experimental


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  Author: Bill Naylor

This CD contains a symbol to represent the generalised hyperbolic function, and facts relating it to other functions.


generalised_hyperbolic

Description:

This symbol represents the generalised hyperbolic function as recorded by Riccati. It is intended to be applied in the curried form, that is, the symbol should be applied to three arguments in order to return a function which should be applied to one argument. The generalised hyperbolic function may be defined as an infinite sum as in the first CMP/FMP .

Commented Mathematical property (CMP):
for complex \alpha, integral n and r an integer between 0 and r (inclusive) (F^\alpha_{n,r})(x) = \Sigma^\infty_{k=0}{\frac{\alpha^k}{(nk+r)!}x^{nk+r}}
Formal Mathematical property (FMP):
alpha C n Z r [ 0 , n - 1 ] ( generalised_hyperbolic ( alpha , n , r ) ) ( x ) = k = 0 alpha k ( n k + r ) ! x ( n k + r )
Commented Mathematical property (CMP):
for all z \in C F^1_{1,0} (z) = e^z
Formal Mathematical property (FMP):
z . z C ( generalised_hyperbolic ( 1 , 1 , 0 ) ) ( z ) = exp ( z )
Commented Mathematical property (CMP):
for all z \in C F^{-1}_{2,-1} (z) = sin(z)
Formal Mathematical property (FMP):
z . z C ( generalised_hyperbolic ( -1 , 2 , -1 ) ) ( z ) = sin ( z )
Commented Mathematical property (CMP):
for all z \in C F^{-1}_{2,0} (z) = cos(z)
Formal Mathematical property (FMP):
z . z C ( generalised_hyperbolic ( -1 , 2 , 0 ) ) ( z ) = cos ( z )
Commented Mathematical property (CMP):
for all z \in C F^{1}_{2,1} (z) = sinh(z)
Formal Mathematical property (FMP):
z . z C ( generalised_hyperbolic ( 1 , 2 , 1 ) ) ( z ) = sinh ( z )
Commented Mathematical property (CMP):
for all z \in C F^{1}_{2,0} (z) = cosh(z)
Formal Mathematical property (FMP):
z . z C ( generalised_hyperbolic ( 1 , 2 , 0 ) ) ( z ) = cosh ( z )
Signatures:
sts


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