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.
An important example of a function of a quaternion variable is
which rotates the vector part of q by twice the angle represented by u.
The quaternion multiplicative inverse is another fundamental function, but it raises difficult questions such as "What should be?" and "What solves the equation ?"
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 can be proven by taking the basis elements. With them, we have
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. 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
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 by the homography composition
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 map of quaternion algebra is called linear, if following equalities hold
where is real field. Since is linear map of quaternion algebra, then, for any , the map
is linear map. If is identity map (), then, for any , we identify tensor product and the map
For any linear map there exists a tensor , , such that
So we can identify the linear map and the tensor .
The derivative for quaternions
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. Considering of the increment of polynomial function of quaternionic argument shows that the increment is linear map of increment of the argument. From this, a definition can be made:
Continuous map is called differentiable on the set , if, at every point , the increment of the map can be represented as
is linear map of quaternion algebra and is such continuous map that
Linear map is called derivative of the map .
On the quaternions, the derivative may be expressed as
Therefore, the differential of the map may be expressed as /math With Brackets on either side.
The number of terms in the sum will depend on the function f. The expressions are called components of derivative.
The derivative of a quaternionic function holds the following equalities
For the function f(x) = axb, the derivative is
and so the components are:
Similarly, for the function f(x) = x^22, the derivative is
and the components are:
Finally, for the function f(x) = x−1, the derivative is
and the components are:
- 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.
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