Quaternionic analysis

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

:$f\_1(q)\; =\; u\; q\; u^\{-1\}$

which rotates the vector part of q by twice the angle represented by the versor u.

The quaternion multiplicative inverse $f\_2(q)\; =\; q^\{-1\}$ is another fundamental function, but as with other number systems, $f\_2(0)$ and related problems are generally excluded due to the nature of dividing by zero.

Affine transformations of quaternions have the form

:$f\_3(q)\; =\; aq\; +\; b,\; \backslash quad\; a,\; b,\; q\; \backslash in\; \backslash mathbb\{H\}.$

Linear fractional transformations of quaternions can be represented by elements of the matrix ring $M\_2(\backslash mathbb\{H\})$ operating on the projective line over $\backslash mathbb\{H\}$. For instance, the mappings $q\; \backslash mapsto\; u\; q\; v,$ where $u$ and $v$ 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 $f\_4(q)\; =\; -\; \backslash tfrac\{1\}\{2\}\; (q\; +\; iqi\; +\; jqj\; +\; kqk)$

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

:$f\_4(1)\; =\; -\backslash tfrac\{1\}\{2\}(1\; -\; 1\; -\; 1\; -\; 1)\; =\; 1,\; \backslash quad$

f_4(i) = -\tfrac{1}{2}(i - i + i + i) = -i, \quad

f_4(j) = -j, \quad

f_4(k) = -k .

Consequently, since $f\_4$ is linear,

:$f\_4(q)\; =\; f\_4(w\; +\; x\; i\; +\; y\; j\; +\; z\; k)\; =\; w\; f\_4(1)\; +\; x\; f\_4(i)\; +\; y\; f\_4(j)\; +\; z\; f\_4(k)\; =\; w\; -\; x\; i\; -\; y\; j\; -\; z\; k\; =\; q^*.$

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.{{harv|Fueter|1936}} These efforts were summarized in {{harvtxt|Deavours|1973}}.{{efn|{{harvtxt|Deavours|1973}} recalls a 1935 issue of Commentarii Mathematici Helvetici where an alternative theory of "regular functions" was initiated by {{harvtxt|Fueter|1936}} through the idea of Morera's theorem: quaternion function $F$ is "left regular at $q$" when the integral of $F$ vanishes over any sufficiently small hypersurface containing $q$. Then the analogue of Liouville's theorem holds: The only regular quaternion function with bounded norm in $\backslash mathbb\{E\}^4$ 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.}}

Though $\backslash mathbb\{H\}$ appears as a union of complex planes, the following proposition shows that extending complex functions requires special care:

Let $f\_5(z)\; =\; u(x,y)\; +\; i\; v(x,y)$ be a function of a complex variable, $z\; =\; x\; +\; i\; y$. Suppose also that $u$ is an even function of $y$ and that $v$ is an odd function of $y$. Then $f\_5(q)\; =\; u(x,y)\; +\; rv(x,y)$ is an extension of $f\_5$ to a quaternion variable $q\; =\; x\; +\; yr$ where $r^2\; =\; -1$ and $r\; \backslash in\; \backslash mathbb\{H\}$.

Then, let $r^*$ represent the conjugate of $r$, so that $q\; =\; x\; -\; yr^*$. The extension to $\backslash mathbb\{H\}$ will be complete when it is shown that $f\_5(q)\; =\; f\_5(x\; -\; yr^*)$. Indeed, by hypothesis

:$u(x,y)\; =\; u(x,-y),\; \backslash quad\; v(x,y)\; =\; -v(x,-y)\; \backslash quad$ one obtains

:$f\_5(x\; -\; y\; r^*)\; =\; u(x,-y)\; +\; r^*\; v(x,-y)\; =\; u(x,y)\; +\; r\; v(x,y)\; =\; f\_5(q).$

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.{{harv|Cayley|1848|loc=especially page 198}}

In terms of a homography, the rotation is expressed

:

where $u\; =\; \backslash exp(\backslash theta\; r)\; =\; \backslash cos\; \backslash theta\; +\; r\; \backslash sin\; \backslash theta$ is a versor. If p * = −p, then the translation $q\; \backslash mapsto\; q\; +\; p$ 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{{harv|Hamilton|1853|loc=§287 pp. 273,4}} by the homography composition

:

where $z\; =\; u\; s\; -\; s\; u\; =\; \backslash sin\; \backslash theta\; (rs\; -\; sr)\; =\; 2\; t\; \backslash sin\; \backslash theta\; .$

Now in the (s,t)-plane the parameter θ traces out a circle $u^\{-1\}\; z\; =\; u^\{-1\}(2\; t\; \backslash sin\; \backslash theta)\; =\; 2\; \backslash sin\; \backslash theta\; (\; t\; \backslash cos\; \backslash theta\; -\; s\; \backslash sin\; \backslash theta)$ in the half-plane $\backslash lbrace\; wt\; +\; xs\; :\; x\; >\; 0\; \backslash rbrace\; .$

Any p in this half-plane lies on a ray from the origin through the circle and can be written $p\; =\; a\; u^\{-1\}\; z\; ,\; \backslash \; \backslash \; a\; >\; 0\; .$

Then up = az, with 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 $\backslash \; f(q)\; =\; q^2\backslash$ from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.{{harvp|Hamilton|1866|loc=Chapter II, On differentials and developments of functions of quaternions, pp. 391–495 }}{{harvp|Laisant|1881|loc=Chapitre 5: Différentiation des Quaternions, pp. 104–117 }}

Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument.{{Dubious|date=March 2019}} From this, a definition can be made:

A continuous function

$\backslash \; f:\; \backslash mathbb\; H\; \backslash rightarrow\; \backslash mathbb\; H\backslash$

is called differentiable on the set $\backslash \; U\; \backslash subset\; \backslash mathbb\; H\backslash \; ,$ if at every point $\backslash \; x\; \backslash in\; U\backslash \; ,$ an increment of the function $\backslash \; f\backslash$ corresponding to a quaternion increment $\backslash \; h\backslash$ of its argument, can be represented as

: $f(x+h)\; -\; f(x)\; =\; \backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\; \backslash circ\; h\; +\; o(h)$

where

: $\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}:\backslash mathbb\; H\backslash rightarrow\backslash mathbb\; H$

is linear map of quaternion algebra $\backslash \; \backslash mathbb\; H\backslash \; ,$ and

$\backslash \; o:\backslash mathbb\; H\backslash rightarrow\; \backslash mathbb\; H\backslash$

represents some continuous map such that

: $\backslash lim\_\{a\; \backslash rightarrow\; 0\}\; \backslash frac\{\backslash \; \backslash left|\backslash \; o(a)\backslash \; \backslash right|\backslash \; \}\{\; \backslash left|\backslash \; a\backslash \; \backslash right|\; \}\; =\; 0\backslash \; ,$

and the notation $\backslash \; \backslash circ\; h\backslash$ denotes ...{{explain|date=September 2024}}

The linear map

$\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}$

is called the derivative of the map $\backslash \; f\; ~.$

On the quaternions, the derivative may be expressed as

: $\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\; =\; \backslash sum\_s\; \backslash frac\{\backslash operatorname\{d\}\_\{s0\}\; f(x)\}\{\backslash operatorname\; d\; x\}\; \backslash otimes\; \backslash frac\{\backslash operatorname\{d\}\_\{s1\}\; f(x)\}\{\backslash operatorname\; d\; x\}$

Therefore, the differential of the map $\backslash \; f\backslash$ may be expressed as follows, with brackets on either side.

:$\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash circ\; \backslash operatorname\; d\; x\; =\; \backslash left(\backslash sum\_s\; \backslash frac\{\backslash operatorname\{d\}\_\{s0\}\; f(x)\}\{\backslash operatorname\; d\; x\}\; \backslash otimes\; \backslash frac\{\backslash operatorname\{d\}\_\{s1\}\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash right)\backslash circ\; \backslash operatorname\; d\; x\; =\; \backslash sum\_s\; \backslash frac\{\backslash operatorname\{d\}\_\{s0\}\; f(x)\}\{\backslash operatorname\; d\; x\}\; \backslash left(\; \backslash operatorname\; d\; x\; \backslash right)\; \backslash frac\{\backslash operatorname\{d\}\_\{s1\}\; f(x)\}\{\backslash operatorname\; d\; x\}$

The number of terms in the sum will depend on the function $\backslash \; f\; ~.$ The expressions

$~~\; \backslash frac\{\backslash operatorname\{d\}\_\{sp\}\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\; ~~\; \backslash mathsf\{\backslash \; for\backslash \; \}\; ~~\; p\; =\; 0,\; 1\; ~~$ are called

components of derivative.

The derivative of a quaternionic function is defined by the expression

: $\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash circ\; h\; =\; \backslash lim\_\{t\backslash to\; 0\}\backslash left(\backslash \; \backslash frac\{\backslash \; f(x\; +\; t\backslash \; h)\; -\; f(x)\backslash \; \}\{\; t\; \}\backslash \; \backslash right)$

where the variable $\backslash \; t\backslash$ is a real scalar.

The following equations then hold:

: $\backslash frac\{\backslash operatorname\; d\backslash left(\; f(x)\; +\; g(x)\; \backslash right)\}\{\backslash operatorname\; d\; x\}\; =\; \backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\; +\; \backslash frac\{\backslash operatorname\; d\; g(x)\}\{\backslash operatorname\; d\; x\}$

: $\backslash frac\{\backslash operatorname\; d\backslash left(\; f(x)\backslash \; g(x)\backslash right)\}\{\backslash operatorname\; d\; x\}\; =\; \backslash frac\{\backslash operatorname\; d\; f(x)\; \}\{\backslash operatorname\; d\; x\}\backslash \; g(x)\; +\; f(x)\backslash \; \backslash frac\{\backslash operatorname\; d\; g(x)\}\{\backslash operatorname\; d\; x\}$

: $\backslash frac\{\backslash operatorname\; d\; \backslash left(\; f(x)\backslash \; g(x)\backslash right)\}\{\backslash operatorname\; d\; x\}\; \backslash circ\; h\; =\; \backslash left(\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash circ\; h\backslash right\; )\backslash \; g(x)\; +\; f(x)\; \backslash left(\backslash frac\{\backslash operatorname\; d\; g(x)\}\{\backslash operatorname\; d\; x\}\backslash circ\; h\backslash right)$

: $\backslash frac\{\backslash operatorname\; d\; \backslash left(\; a\backslash \; f(x)\backslash \; b\backslash right)\}\{\backslash operatorname\; d\; x\}\; =\; a\backslash \; \backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash \; b$

: $\backslash frac\{\backslash operatorname\; d\; \backslash left(\; a\backslash \; f(x)\backslash \; b\backslash right)\}\{\backslash operatorname\; d\; x\}\backslash circ\; h\; =\; a\; \backslash left(\backslash frac\{\backslash operatorname\; d\; f(x)\}\{\backslash operatorname\; d\; x\}\backslash circ\; h\backslash right)\; b$

For the function $\backslash \; f(x)\; =\; a\backslash \; x\backslash \; b\backslash \; ,$ where $\backslash \; a\backslash$ and $\backslash \; b\backslash$ are constant quaternions, the derivative is

and so the components are:

Similarly, for the function $\backslash \; f(x)\; =\; x^2\backslash \; ,$ the derivative is

and the components are:

Finally, for the function $\backslash \; f(x)\; =\; x^\{-1\}\backslash \; ,$ the derivative is

and the components are:

• Cayley transform
• Quaternionic manifold

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