...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 quaternions 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 quaternions ("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.
This author claims the complete proof of the "closed formula" as
his own, as well as its independant invention (there are claims to prior invention
of the formula, such as one by Professor Shoemake, and it is possible that
the formula had been known a couple of centuries back, but in absence of bibliographical
reference, the matter is pending, awaiting further investigation; on the other
hand, the definition and existence of the exponential on the quaternions, is
of course a fact known for a very long time). Basically, any converging power
series with real coefficients which allows for a closed formula in * C* can be transposed to

template<typename T> quaternion<T> exp(quaternion<T> const & q);

Computes the exponential of the quaternion.

template<typename T> quaternion<T> cos(quaternion<T> const & q);

Computes the cosine of the quaternion

template<typename T> quaternion<T> sin(quaternion<T> const & q);

Computes the sine of the quaternion.

template<typename T> quaternion<T> tan(quaternion<T> const & q);

Computes the tangent of the quaternion.

template<typename T> quaternion<T> cosh(quaternion<T> const & q);

Computes the hyperbolic cosine of the quaternion.

template<typename T> quaternion<T> sinh(quaternion<T> const & q);

Computes the hyperbolic sine of the quaternion.

template<typename T> quaternion<T> tanh(quaternion<T> const & q);

Computes the hyperbolic tangent of the quaternion.

template<typename T> quaternion<T> pow(quaternion<T> const & q, int n);

Computes the n-th power of the quaternion q.