This document can be viewed properly only in modern browsers with good support for CSS2 and the full HTML character set.

Quaternion Analysis
Functions of a Quaternion Variable

Prerequisites

To understand this, you need a knowledge of complex analysis, such as is taught in third or fourth year college math courses, and some understanding of divergence-type theorems, as presented in college advanced calculus courses.

This is a work in progress. Have a look, but don’t take it too seriously.

I have done my best to survey the results in this field. My primary reference in all of this was the excellent 1979 paper by Anthony Sudbery. This paper is written at a level that isn’t approachable by most physicists or undergraduate math students. The other major papers in the field are those by Fueter et al. from the 1930s; these lay the groundwork for calculus and analysis of quaternion functions, and draw several important parallels with the theory of functions of a complex variable.

Definition of Quaternions

See A definition of the quaternions for a working definition.

The algebra of quaternions is usually denoted by the letter ℍ (for their inventor, Hamilton).

Quaternion analysis vs. complex analysis

In the following, the term quaternion function is used to mean a quaternion-valued function of one quaternion variable, and complex function to mean a complex-valued function of one complex variable.

The analysis of quaternion functions is much more complicated than the analysis of complex functions, as is to be expected. It is marked from the beginning by the left-right multiplicative dichotomy.

The first surprise may not be pleasant. The synonymity in complex analysis of the four notions:

• holomorphy: existence of a difference quotient,
• harmonicity: satisfaction of Laplace’s equation,
• analyticity: expressibilty in terms of power series, and
• conformality: local preservation of geometry

does not hold for quaternion functions. For quaternions, each of these must be re-evaluated; nothing survives without alteration.

Most of the beautiful results in the theory of complex functions have analogs in the theory of quaternion functions, however.

Regularity of quaternion functions

Because of the non-commutativity of quaternions, the meaning of quaternion power series is unclear, and the most obvious generalizations of the difference quotient are useless.

The geometrical notion of conformality extends to the quaternions, bringing with it the Cauchy formulas and much of the beauty of the analysis of the complex functions. It can be characterized by an extension of the Cauchy-Riemann equations.

Difference Quotient

Since the quaternions possess a multiplicative inverse, one is compelled to experiment on functions of quaternions with the difference quotient ( f( x + h ) − f( x ) ) / h that works so well with functions of real and complex numbers.

An issue with commutativity arises immediately: there would be at least a left and a right difference quotient. The quotient could be replaced by a multiplication on the left and right by h⁻¹, respectively.

It turns out that the only quaternion functions of x for which the limit of a right difference quotient exist are of the form p + qx, and those for which the limit of a left difference quotient exist are of the form p + xq.

The requirement that the limit of the obvious difference quotient should exist is too strong a condition to be useful for the quaternions.

A more elaborate difference quotient can be used to represent a kind of derivative for quaternions. Sudbery introduces

This is a quaternion parallel to a, b and c whose length is the area of the three-parallelepiped determined by a, b and c. (Compare with the cross product of three-vectors.)

With this and the divergence theorem, he writes a left derivative as

where a, b, and c are real multiples of any three fixed linearly independent quaternions, and the limit is taken as these multiples go to zero.

Analyticity

A function is analytic at a point if in a neighborhood of that point, it is the limit of a power series.

Once again, commutativity is a problem. If one allows terms with coefficients in any order, such as q₁ x q₂ x q₃ for the second-order term, it turns out the theory is the same as that of Fréchet-differentiable functions from ℝ⁴ into itself. But since complex analyticity is a much stronger property than Fréchet-differentiability—something seems to have been lost.

The reason for this is that, in contrast to the case for complex numbers, each real component of a quaternion can be expressed as a polynomial in the quaternion. For example, if q is a quaternion and q = t + x i + y j + z k, where t, x, y, z are real, then

That is, any polynomial in t, x, y, z can be expressed as a quaternion polynomial (at least one with the general coefficients described above).

So, the existence of a power series representation with general quaternion coefficients is too weak a condition to be useful.

Things get worse still for the general theory of power series, The quaternion function q² is not quaternion-differentiable in any sense known to the author. This destroys much of the usefulness of power series: one can’t write the derivative of the series as the series of derivatives.

This perplexing issue has several aspects. We have all been drilled in the use of polynomials and power series. But what is the real worth of power series in analysis? Is it possible to recover from the loss of power series?

Any complex-analytic function can be extended to the quaternions as described below. This extension also works for polynomials, but the extension is not a quaternion polynomial.

What are the practical consequences of the lack of power series representations of quaternion functions? One important use of power series is to reduce the evaluation of a transcendental function to arithmetic computation. It can be shown that, with a natural definition of regularity, a regular quaternion function is infinitely real-differentiable. The function therefore has a representation as a power series in the four real components of its argument. This could take the place the computational use of quaternion power series.

differentiability conditions

One definition of regularity for quaternion functions comes from an extension of the Cauchy-Riemann equations. Let u and v be complex-valued functions each of two complex variables x and y then (understanding the multiplication to follow formally the formula for quaternions),

is the condition¹ for a quaternion function f = (u, v) to be called left-regular. To make the notation work neatly for the analogous condition for f to be right-regular, we’ll write a partial derivative operator that is applied to the function on its left as ·∂/∂x. The condition for right-regularity can be written

The identity function is regular by this definition, but no higher order polynomial is.

Cauchy’s theorem

The quaternion analog of Cauchy’s theorem involves integration over a three-dimensional manifold in the four-dimensional quaternion space, rather than a one-dimensional curve in the two-dimensional complex plane. It’s not a surprise if you’re familiar with differential forms and the generalized divergence theorem.

Special functions

The real and complex versions of the exponential function e^z serve as isomorphisms between their respective additive and multiplicative groups by way of the formula

There can be no such formula in the quaternion algebra, because its additive and multiplicative groups are not homomorphic: its additive group is commutative while its multiplicative group is not. The trigonometric and logarithmic functions are beset with corresponding problems.

What are the alternatives?

One is to extend the complex exponential function to a regular quaternion function. This function might have other nice properties, but it won’t have the above formula, which was formerly the definitive property of exponentials.

Another is to define the exponential of a quaternion variable to be a mapping to some other algebra that is commutative.

Extension of complex functions to quaternions

Cauchy’s integral formula

Stokes’ theorem in the plane involves an oriented integration over a closed curve (a one-manifold in ℝ²). The version in ℝ⁴ involves an oriented three-manifold.

The elementary potential function for ℝ⁴ is written in quaternion form as

Its Fréchet derivative, also written in quaternion notation, is

Now the Cauchy-Fueter integral formula can be restated as in Sudbery, where it is proved for a special case of a four-parallelepiped.

If f is regular at every point of a positively oriented four-parallelepiped C, and q₀ is a point in the interior of C,

Conformality

In complex analysis, conformality of a function is its property of preserving local geometry, in the sense that as a triangle is shrunk isometrically to 0, the image of the triangle under the function is asymptotically similar to the original. In the limit, the angles of the original and image triangles are the same, and have the same order, although the image triangle may be larger or smaller than the original, and rotated.

Another way of saying this is that locally, a conformal function has the same effect as multiplying by a complex number (namely, the derivative of the function).

There are two other kinds of linear deformations in the plane that cannot be achieved by multiplying by a complex number: reflection and skew. A skew transformation doesn't preserve angle, and a reflection of the plane reverses the sense of traversal of the vertices of a triangle.

The left- and right-regularity of a quaternion function mean that locally, the function has the effect of multiplying on the left or right, respectively, by a quaternion (the left and right derivatives). One would expect that such transformations preserve something of the geometry of four space.

Just what transformations of four space are effected by quaternion multiplication? Also, what is the geometrical distinction between left and right multiplication? (I don't know the answers: work in progress)

The conformality of quaternion multiplication is quite analogous to that of complex multiplication. The angles (two, three, and four-dimensional) are preserved, along with the senses. As in the two-dimensional case, this precludes transformations that effect flips of coordinates.

However, there is a new flavor to the four dimensional conformal transformations absent in two or three dimensions: a twisting (or handedness, or chirality) that is essentially either of two directions. And the handedness of the quaternion multiplication corresponds to the handedness of the transformation.

Looking at quaternion multiplication as a transformation of three-dimensional orientation, there are two essentially different kinds of direct paths from one orientation to another: one which twists to the left, one to the right. (Often one path is much shorter than the other.)

From the point of view of an observer strapped into a chair, being moved from one orientation to another, the two motions also have a distinct feeling. (Indeed, some interactive 3-D video games make use of quaternion transformations of the viewer's orientation in a virtual world, but some use only quaternion multiplication from a single side: the effect is an unnatural sensation of always twisting the same direction from one view to another.)

So quaternion transformations come in two distinct kinds, as do transformations of three-dimensional orientations. These transformations are also distant from one another, in the sense of the usual measure of distance between linear transformations of four space. The transformation effected by left multiplication by a non-scalar quaternion is arbitrarily near transformations effected by other left quaternion multiplications, but it is distant from that of any right quaternion multiplication by some fixed minimum positive number. Only the transformations effected by scalar quaternions are arbitrarily near those of both left and right quaternion transformations.

Viewed as a subset of the linear space of transformations of four space, the transformations effected by left and right multiplication by quaternions constitute two four-dimensional subspaces, intersecting along the one dimension of isometric scalings of four space (multiplication by a scalar quaternion).

Notes

¹ Since the conditions were first published by Fueter, and since they are analogous to the differentiability conditions for complex functions which are called Cauchy-Riemann equations, these equations for quaternion functions are often called the Cauchy-Riemann-Fueter equations.

See also