...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

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

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.