As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions.
Unlike the complex numbers and like the reals, the four notions do not coincide.
The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure.
An important example of a function of a quaternion variable is which rotates the vector part of q by twice the angle represented by the versor u.
is another fundamental function, but as with other number systems,
and related problems are generally excluded due to the nature of dividing by zero.
Affine transformations of quaternions have the form Linear fractional transformations of quaternions can be represented by elements of the matrix ring
are fixed versors serve to produce the motions of elliptic space.
For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation.
This equation can be proven, starting with the basis {1, i, j, k}: Consequently, since
is linear, The success of complex analysis in providing a rich family of holomorphic functions for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable.
appears as a union of complex planes, the following proposition shows that extending complex functions requires special care: Let
Indeed, by hypothesis In the following, colons and square brackets are used to denote homogeneous vectors.
The rotation about axis r is a classical application of quaternions to space mapping.
is expressed by Rotation and translation xr along the axis of rotation is given by Such a mapping is called a screw displacement.
In classical kinematics, Chasles' theorem states that any rigid body motion can be displayed as a screw displacement.
Just as the representation of a Euclidean plane isometry as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the screw axis required, is a matter of quaternion arithmetic with homographies: Let s be a right versor, or square root of minus one, perpendicular to r, with t = rs.
Consider the axis passing through s and parallel to r. Rotation about it is expressed[3] by the homography composition where
Now in the (s,t)-plane the parameter θ traces out a circle
Any p in this half-plane lies on a ray from the origin through the circle
as the homography expressing conjugation of a rotation by a translation p. Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even
Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.
[dubious – discuss] From this, a definition can be made: A continuous function
of its argument, can be represented as where is linear map of quaternion algebra
denotes ...[further explanation needed] The linear map
On the quaternions, the derivative may be expressed as Therefore, the differential of the map
The number of terms in the sum will depend on the function
The derivative of a quaternionic function is defined by the expression where the variable
are constant quaternions, the derivative is and so the components are: Similarly, for the function
the derivative is and the components are: Finally, for the function