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Octonions Transcendentals

There is no log or sqrt provided for octonions in this implementation, and pow is likewise restricted to integral powers of the exponent. There are several reasons to this: on the one hand, the equivalent of analytic continuation for octonions ("branch cuts") remains to be investigated thoroughly (by me, at any rate...), and we wish to avoid the nonsense introduced in the standard by exponentiations of complexes by complexes (which is well defined, but not in the standard...). Talking of nonsense, saying that pow(0,0) is "implementation defined" is just plain brain-dead...

We do, however provide several transcendentals, chief among which is the exponential. That it allows for a "closed formula" is a result of the author (the existence and definition of the exponential, on the octonions among others, on the other hand, is a few centuries old). Basically, any converging power series with real coefficients which allows for a closed formula in C can be transposed to O. More transcendentals of this type could be added in a further revision upon request. It should be noted that it is these functions which force the dependency upon the boost/math/special_functions/sinc.hpp and the boost/math/special_functions/sinhc.hpp headers.

template<typename T>
octonion<T> exp(octonion<T> const & o);

Computes the exponential of the octonion.

template<typename T>
octonion<T> cos(octonion<T> const & o);

Computes the cosine of the octonion

template<typename T>
octonion<T> sin(octonion<T> const & o);

Computes the sine of the octonion.

template<typename T>
octonion<T> tan(octonion<T> const & o);

Computes the tangent of the octonion.

template<typename T>
octonion<T> cosh(octonion<T> const & o);

Computes the hyperbolic cosine of the octonion.

template<typename T>
octonion<T> sinh(octonion<T> const & o);

Computes the hyperbolic sine of the octonion.

template<typename T>
octonion<T> tanh(octonion<T> const & o);

Computes the hyperbolic tangent of the octonion.

template<typename T>
octonion<T>  pow(octonion<T> const & o, int n);

Computes the n-th power of the octonion q.