gen_hyperbolic1 http://www.openmath.org/cd/gen_hyperbolic1.ocd 2005-01-01 experimental 2002-11-11 0 0 alg1 arith1 fns1 integer1 interval1 logic1 nums1 quant1 relation1 set1 setname1 transc1 This CD contains a symbol to represent the generalised hyperbolic function, and facts relating it to other functions. generalised_hyperbolic 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 . 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}} 0 1 for all z \in C F^1_{1,0} (z) = e^z 1 1 0 for all z \in C F^{-1}_{2,-1} (z) = sin(z) -1 2 -1 for all z \in C F^{-1}_{2,0} (z) = cos(z) -1 2 0 for all z \in C F^{1}_{2,1} (z) = sinh(z) 1 2 1 for all z \in C F^{1}_{2,0} (z) = cosh(z) 1 2 0