Inmathematics,quaternionic analysis is the study offunctions withquaternions as thedomain and/or range. Such functions can be calledfunctions of a quaternion variable just asfunctions of a real variable or acomplex variable are called.

As withcomplex andreal analysis, it is possible to study the concepts ofanalyticity,holomorphy,harmonicity andconformality in the context of quaternions. Unlike thecomplex numbers and like thereals, the four notions do not coincide.

Theprojections of a quaternion onto its scalar part or onto its vector part, as well as the modulus andversor functions, are examples that are basic to understanding quaternion structure.

An important example of a function of a quaternion variable is

${\displaystyle f_1(q)=uq{u}^{-1}}$

whichrotates the vector part ofq by twice the angle represented by the versoru.

The quaternionmultiplicative inverse${\displaystyle f_2(q)=q^{-1}}$ is another fundamental function, but as with other number systems,${\displaystyle f_2(0)}$ and related problems are generally excluded due to the nature ofdividing by zero.

Affine transformations of quaternions have the form

${\displaystyle f_3(q)=aq+b,a,b,q\in \mathbb{H}.}$

Linear fractional transformations of quaternions can be represented by elements of thematrix ring${\displaystyle M_2(\mathbb{H})}$ operating on theprojective line over${\displaystyle \mathbb{H}}$. For instance, the mappings${\displaystyle q\mapsto uqv,}$ where${\displaystyle u}$ and${\displaystyle v}$ are fixedversors serve to produce themotions of elliptic space.

Quaternion variable theory differs in some respects from complex variable theory. For example: Thecomplex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic,non-analytic operation. Indeed, conjugation changes theorientation of plane figures, something that arithmetic functions do not change.

In contrast to thecomplex conjugate, the quaternion conjugation can be expressed arithmetically, as${\displaystyle f_4(q)=-\frac{1}{2}(q+iqi+jqj+kqk)}$

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

${\displaystyle f_4(1)=-\frac{1}{2}(1-1-1-1)=1,f_4(i)=-\frac{1}{2}(i-i+i+i)=-i,f_4(j)=-j,f_4(k)=-k}$.

Consequently, since${\displaystyle f_4}$ islinear,

${\displaystyle f_4(q)=f_4(w+xi+yj+zk)=w{f}_4(1)+x{f}_4(i)+y{f}_4(j)+z{f}_4(k)=w-xi-yj-zk=q^{\ast }.}$

The success ofcomplex analysis in providing a rich family ofholomorphic 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 inDeavours (1973).^[a]

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

Let${\displaystyle f_5(z)=u(x,y)+iv(x,y)}$ be a function of a complex variable,${\displaystyle z=x+iy}$. Suppose also that${\displaystyle u}$ is aneven function of${\displaystyle y}$ and that${\displaystyle v}$ is anodd function of${\displaystyle y}$. Then${\displaystyle f_5(q)=u(x,y)+rv(x,y)}$ is an extension of${\displaystyle f_5}$ to a quaternion variable${\displaystyle q=x+yr}$ where${\displaystyle r^2=-1}$ and${\displaystyle r\in \mathbb{H}}$.Then, let${\displaystyle r^{\ast }}$ represent the conjugate of${\displaystyle r}$, so that${\displaystyle q=x-y{r}^{\ast }}$. The extension to${\displaystyle \mathbb{H}}$ will be complete when it is shown that${\displaystyle f_5(q)=f_5(x-y{r}^{\ast })}$. Indeed, by hypothesis

${\displaystyle u(x,y)=u(x,-y),v(x,y)=-v(x,-y)}$ one obtains

${\displaystyle f_5(x-y{r}^{\ast })=u(x,-y)+r^{\ast }v(x,-y)=u(x,y)+rv(x,y)=f_5(q).}$

In the following, colons and square brackets are used to denotehomogeneous vectors.

Therotation about axisr is a classical application of quaternions tospace mapping.^[2]In terms of ahomography, the rotation is expressed

where${\displaystyle u=\exp (\theta r)=\cos \theta +r\sin \theta }$ is aversor. Ifp * = −p, then the translation${\displaystyle q\mapsto q+p}$ is expressed by

Rotation and translationxr along the axis of rotation is given by

Such a mapping is called ascrew displacement. In classicalkinematics,Chasles' theorem states that any rigid body motion can be displayed as a screw displacement. Just as the representation of aEuclidean plane isometry as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and thescrew axis required, is a matter of quaternion arithmetic with homographies: Lets be a right versor, or square root of minus one, perpendicular tor, witht =rs.

Consider the axis passing throughs and parallel tor. Rotation about it is expressed^[3] by the homography composition

where${\displaystyle z=us-su=\sin \theta (rs-sr)=2t\sin \theta .}$

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

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

Thenup =az, with as the homography expressingconjugation of a rotation by a translation p.

Since the time of Hamilton, it has been realized that requiring the independence of thederivative from the path that a differential follows toward zero is too restrictive: it excludes even${\displaystyle f(q)=q^2}$ from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.^[4][5]Considering the increment ofpolynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument.^{[dubious –discuss]} From this, a definition can be made:

A continuous function${\displaystyle f:\mathbb{H}\to \mathbb{H}}$is calleddifferentiable on the set${\displaystyle U\subset \mathbb{H},}$ if at every point${\displaystyle x\in U,}$ an increment of the function${\displaystyle f}$ corresponding to a quaternion increment${\displaystyle h}$ of its argument, can be represented as

${\displaystyle f(x+h)-f(x)=\frac{\mathrm{d}f(x)}{\mathrm{d}x}\circ h+o(h)}$

where

${\displaystyle \frac{\mathrm{d}f(x)}{\mathrm{d}x}:\mathbb{H}\to \mathbb{H}}$

islinear map of quaternion algebra${\displaystyle \mathbb{H},}$ and${\displaystyle o:\mathbb{H}\to \mathbb{H}}$represents some continuous map such that

${\displaystyle \underset{a\to 0}{lim}\frac{\left|o(a)\right|}{\left|a\right|}=0,}$

and the notation${\displaystyle \circ h}$ denotes ...^{[further explanation needed]}

The linear map${\displaystyle \frac{\mathrm{d}f(x)}{\mathrm{d}x}}$is called the derivative of the map${\displaystyle f.}$

On the quaternions, the derivative may be expressed as

${\displaystyle \frac{\mathrm{d}f(x)}{\mathrm{d}x}=\sum _s\frac{{\mathrm{d}}_{s0}f(x)}{\mathrm{d}x}\otimes \frac{{\mathrm{d}}_{s1}f(x)}{\mathrm{d}x}}$

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

${\displaystyle \frac{\mathrm{d}f(x)}{\mathrm{d}x}\circ \mathrm{d}x=\left(\sum _s\frac{{\mathrm{d}}_{s0}f(x)}{\mathrm{d}x}\otimes \frac{{\mathrm{d}}_{s1}f(x)}{\mathrm{d}x}\right)\circ \mathrm{d}x=\sum _s\frac{{\mathrm{d}}_{s0}f(x)}{\mathrm{d}x}\left(\mathrm{d}x\right)\frac{{\mathrm{d}}_{s1}f(x)}{\mathrm{d}x}}$

The number of terms in the sum will depend on the function${\displaystyle f.}$ The expressions${\displaystyle \frac{{\mathrm{d}}_{sp}\mathrm{d}f(x)}{\mathrm{d}x}\mathsf{f}\mathsf{o}\mathsf{r}p=0,1}$ are calledcomponents of derivative.

The derivative of a quaternionic function is defined by the expression

${\displaystyle \frac{\mathrm{d}f(x)}{\mathrm{d}x}\circ h=\underset{t\to 0}{lim}\left(\frac{f(x+th)-f(x)}{t}\right)}$

where the variable${\displaystyle t}$ is a real scalar.

The following equations then hold:

${\displaystyle \frac{\mathrm{d}\left(f(x)+g(x)\right)}{\mathrm{d}x}=\frac{\mathrm{d}f(x)}{\mathrm{d}x}+\frac{\mathrm{d}g(x)}{\mathrm{d}x}}$

${\displaystyle \frac{\mathrm{d}\left(f(x)g(x)\right)}{\mathrm{d}x}=\frac{\mathrm{d}f(x)}{\mathrm{d}x}g(x)+f(x)\frac{\mathrm{d}g(x)}{\mathrm{d}x}}$

${\displaystyle \frac{\mathrm{d}\left(f(x)g(x)\right)}{\mathrm{d}x}\circ h=\left(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\circ h\right)g(x)+f(x)\left(\frac{\mathrm{d}g(x)}{\mathrm{d}x}\circ h\right)}$

${\displaystyle \frac{\mathrm{d}\left(af(x)b\right)}{\mathrm{d}x}=a\frac{\mathrm{d}f(x)}{\mathrm{d}x}b}$

${\displaystyle \frac{\mathrm{d}\left(af(x)b\right)}{\mathrm{d}x}\circ h=a\left(\frac{\mathrm{d}f(x)}{\mathrm{d}x}\circ h\right)b}$

For the function${\displaystyle f(x)=axb,}$ where${\displaystyle a}$ and${\displaystyle b}$ are constant quaternions, the derivative is

${\displaystyle \frac{\mathrm{d}\left(axb\right)}{\mathrm{d}x}=a\otimes b}$

${\displaystyle \mathrm{d}y=\frac{\mathrm{d}\left(axb\right)}{\mathrm{d}x}\circ \mathrm{d}x=a\left(\mathrm{d}x\right)b}$

and so the components are:

${\displaystyle \frac{{\mathrm{d}}_{10}\left(axb\right)}{\mathrm{d}x}=a}$

${\displaystyle \frac{{\mathrm{d}}_{11}\left(axb\right)}{\mathrm{d}x}=b}$

Similarly, for the function${\displaystyle f(x)=x^2,}$ the derivative is

${\displaystyle \frac{\mathrm{d}{x}^2}{\mathrm{d}x}=x\otimes 1+1\otimes x}$

${\displaystyle \mathrm{d}y=\frac{\mathrm{d}{x}^2}{\mathrm{d}x}\circ \mathrm{d}x=x\mathrm{d}x+(\mathrm{d}x)x}$

and the components are:

${\displaystyle \frac{{\mathrm{d}}_{10}{x}^2}{\mathrm{d}x}=x}$		${\displaystyle \frac{{\mathrm{d}}_{11}{x}^2}{\mathrm{d}x}=1}$
${\displaystyle \frac{{\mathrm{d}}_{20}{x}^2}{\mathrm{d}x}=1}$		${\displaystyle \frac{{\mathrm{d}}_{21}{x}^2}{\mathrm{d}x}=x}$

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

${\displaystyle \frac{\mathrm{d}{x}^{-1}}{\mathrm{d}x}=-x^{-1}\otimes x^{-1}}$

${\displaystyle \mathrm{d}y=\frac{\mathrm{d}{x}^{-1}}{\mathrm{d}x}\circ \mathrm{d}x=-x^{-1}(\mathrm{d}x)x^{-1}}$

and the components are:

${\displaystyle \frac{{\mathrm{d}}_{10}{x}^{-1}}{\mathrm{d}x}=-x^{-1}}$

${\displaystyle \frac{{\mathrm{d}}_{11}{x}^{-1}}{\mathrm{d}x}=x^{-1}}$

• Cayley transform
• Quaternionic manifold

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• Functions and mappings
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