Quaternionic analysis

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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

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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   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   operating on the projective line over  . For instance, the mappings   where   and   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, the quaternion conjugation can be expressed arithmetically, as  

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.[1] These efforts were summarized in Deavours (1973).[a]

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

Let   be a function of a complex variable,  . Suppose also that   is an even function of   and that   is an odd function of  . Then   is an extension of   to a quaternion variable   where   and  . Then, let   represent the conjugate of  , so that  . The extension to   will be complete when it is shown that  . Indeed, by hypothesis

  one obtains
 

Homographies

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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.[2] In terms of a homography, the rotation is expressed

 

where   is a versor. If p * = −p, then the translation   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   in the half-plane  

Any p in this half-plane lies on a ray from the origin through the circle   and can be written  

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

The derivative for quaternions

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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   from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.[4][5] Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument.[dubiousdiscuss] From this, a definition can be made:

A continuous function   is called differentiable on the set   if at every point   an increment of the function   corresponding to a quaternion increment   of its argument, can be represented as

 

where

 

is linear map of quaternion algebra   and   represents some continuous map such that

 

and the notation   denotes ...[further explanation needed]

The linear map   is called the derivative of the map  

On the quaternions, the derivative may be expressed as

 

Therefore, the differential of the map   may be expressed as follows, with brackets on either side.

 

The number of terms in the sum will depend on the function   The expressions   are called components of derivative.

The derivative of a quaternionic function is defined by the expression

 

where the variable   is a real scalar.

The following equations then hold:

 
 
 
 
 

For the function   where   and   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   the derivative is

   

and the components are:

   

See also

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Notes

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  1. ^ Deavours (1973) recalls a 1935 issue of Commentarii Mathematici Helvetici where an alternative theory of "regular functions" was initiated by Fueter (1936) through the idea of Morera's theorem: quaternion function   is "left regular at  " when the integral of   vanishes over any sufficiently small hypersurface containing  . Then the analogue of Liouville's theorem holds: The only regular quaternion function with bounded norm in   is a constant. One approach to construct regular functions is to use power series with real coefficients. Deavours also gives analogues for the Poisson integral, the Cauchy integral formula, and the presentation of Maxwell’s equations of electromagnetism with quaternion functions.

Citations

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  1. ^ (Fueter 1936)
  2. ^ (Cayley 1848, especially page 198)
  3. ^ (Hamilton 1853, §287 pp. 273,4)
  4. ^ Hamilton (1866), Chapter II, On differentials and developments of functions of quaternions, pp. 391–495
  5. ^ Laisant (1881), Chapitre 5: Différentiation des Quaternions, pp. 104–117

References

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