In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.

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.

Properties

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.

The quaternion multiplicative inverse 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 operating on the \mathbb{H} are fixed versors serve to produce the motions of elliptic space.

Quaternion variable theory differs in some respects from complex variable theory. 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. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change.

In contrast to the complex conjugate

This equation can be proven, starting with the basis {1, i, j, k}:

1. .

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. These efforts were summarized in .

appears as a union of complex planes, the following proposition shows that extending complex functions requires special care:

is an even function is an odd function. Indeed, by hypothesis

1. one obtains

Homographies

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. In terms of a homography, the rotation is expressed

is a versor. If p * = p 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 by the homography composition

Now in the (s,t)-plane the parameter

Any p

Then up = az as the homography expressing conjugation of a rotation by a translation p.

The derivative for quaternions

Since the time of Hamilton, it has been realized that requiring the independence of the derivative from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable. Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument. From this, a definition can be made:

is called differentiable on the set of its argument, can be represented as

where

is linear map represents some continuous map such that

denotes ...

On the quaternions, the derivative may be expressed as

may be expressed as follows, with brackets on either side.

are called components of derivative.

The derivative of a quaternionic function is defined by the expression

is a real scalar.

The following equations then hold:

are constant quaternions, the derivative isand so the components are: the derivative isand the components are: the derivative isand the components are:

See also

• Cayley transform

• Quaternionic manifold

Notes

Citations

References