Quaternionic analysis: Difference between revisions

Revision as of 03:17, 25 January 2018

Latest revision as of 11:23, 26 February 2025

(48 intermediate revisions by 25 users not shown)

Line 1:

{{Short description|Function theory with quaternion variable}}

In [[mathematics]], '''quaternionic analysis''' is the study of functions with [[quaternion]]s as the ___domain and/or range. Such functions can be called '''functions of a quaternion variable''' just as functions of a [[Function of a real variable|real variable]] or functions of a [[complex variable]] are called.

In [[mathematics]], '''quaternionic analysis''' is the study of [[function (mathematics)|functions]] with [[quaternion]]s as the [[___domain of a function|___domain]] and/or range. Such functions can be called '''functions of a quaternion variable''' just as [[Function of a real variable|functions of a real variable]] or a [[Function of a complex variable|complex variable]] are called.

As with [[complex analysis|complex]] and [[real analysis]], it is possible to study the concepts of [[analytic function|analyticity]], [[holomorphic function|holomorphy]], [[harmonic function|harmonicity]] and [[conformality]] in the context of quaternions. It is known that forUnlike the [[complex numbers, these four notions coincide; however, for the quaternions,number]]s and alsolike the [[real numbersnumber|reals]], not all of the four notions aredo thenot samecoincide.

==DiscussionProperties==

The [[projection (linear algebra)|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

:<math>f_1(q) = u q u^{-1}</math>

which [[quaternions and spatial rotation|rotates the vector part of ''q'']] by twice the angle represented by the versor ''u''.

The quaternion [[multiplicative inverse]] <math>f_2(q) = q^{-1}</math> is another fundamental function, but itas raiseswith difficultother questionsnumber such as "What shouldsystems, <math>f_2(0)</math> be?" and "Whatrelated <math>q</math>problems are generally solvesexcluded due to the equationnature <math>f_2(q)of =[[dividing 0</math>?"by zero]].

[[Affine transformation]]s of quaternions have the form

Line 18 ⟶ 19:

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, [[Holomorphic function|non-analytic]] operation. Indeed, conjugation changes the [[orientation (mathematics)|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 <math>f_4(q) = - \tfrac{1}{2} (q + iqi + jqj + kqk)</math>

This equation can be proven, starting with the [[basis (linear algebra)|basis]] {1, i, j, k}:

'''Proposition:''' The function <math>f_4(q) = - \tfrac{1}{2} (q + iqi + jqj + kqk)</math> is equivalent to quaternion conjugation.

'''Proof''': For the basis elements we have

:<math>f_4(1) = -\tfrac{1}{2}(1 - 1 - 1 - 1) = 1, \quad

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

f_4(j) = -j, \quad

f_4(k) = -k </math>.

Consequently, since <math>f_4</math> is [[linear functionmap|linear]],

:<math>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^*.</math>

An immediate '''corollary''' of which is that the quaternion conjugate is [[Analytic function|analytic]] everywhere in <math>\mathbb{H}.</math> Compare this to the seemingly identical complex conjugate, <math>(x + iy)^* = x - iy,</math> for <math>x, y \in \mathbb{R},</math> and <math>i^2 = -1,</math> which is not [[Holomorphic function|analytic]] in <math>\mathbb{C}</math>.

The success of [[complex analysis]] in providing a rich family of [[holomorphic function]]s 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.<ref name=Fueter_1936>[[Rudolf Fueter]] (1936) "Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen" (in German), ''[[Commentarii Mathematici Helvetici]]'' 8: 371–378</ref> These efforts were summarized in 1973 by C.A. Deavours.<ref name=Deavours_1973>C.A. Deavours (1973) "The Quaternion Calculus", ''[[American Mathematical Monthly]]'' 80:995–1008.</ref>{{efn|Devours recalls<ref name=Deavours_1973/> a 1935 issue of ''[[Commentarii Mathematici Helvetici]]'' where an alternative theory of "regular functions" was initiated by [[Rudolf Fueter|R. Fueter]]<ref name=Fueter_1936/> through the idea of [[Morera's theorem]]: quaternion function <math>F</math> is "left regular at <math>q</math>" when the integral of <math>F</math> vanishes over any sufficiently small [[hypersurface]] containing <math>q</math>. Then the analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] holds: The only regular quaternion function with bounded norm in <math>\mathbb{E}^4</math> 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.}}

The success of [[complex analysis]] in providing a rich family of [[holomorphic function]]s 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.<ref>{{harv|Fueter|1936}}</ref> 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 <math>F</math> is "left regular at <math>q</math>" when the integral of <math>F</math> vanishes over any sufficiently small [[hypersurface]] containing <math>q</math>. Then the analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] holds: The only regular quaternion function with bounded norm in <math>\mathbb{E}^4</math> 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 <math>\mathbb{H}</math> [[quaternion#H as a union of complex planes|appears as a union of complex planes]], the following proposition shows that extending complex functions requires special care:

Though <math>\mathbb{H}</math> [[quaternion#As a union of complex planes|appears as a union of complex planes]], the following proposition shows that extending complex functions requires special care:

'''Proposition:''' Let <math>f_5(z) = u(x,y) + i v(x,y)</math> be a function of a complex variable, <math>z = x + i y</math>. Suppose also that <math>u</math> is an [[even function]] of <math>y</math> and that <math>v</math> is an [[odd function]] of <math>y</math>. Then <math>f_5(q) = u(x,y) + rv(x,y)</math> is an extension of <math>f_5</math> to a quaternion variable <math>q = x + yr</math> where <math>r^2 = -1</math> and <math>r \in \mathbb{H}</math>.

'''Proof''': Let <math>r^*f_5(z) = u(x,y) + i v(x,y)</math> representbe thea conjugatefunction of a complex variable, <math>rz = x + i y</math>,. soSuppose also that <math>qu</math> =is xan -[[even function]] of yr^*<math>y</math>. Theand extension tothat <math>\mathbb{H}v</math> willis bean complete[[odd whenfunction]] itof is<math>y</math>. shown thatThen <math>f_5(q) = f_5u(x,y) -+ rv(x,y)</math> is an extension of <math>f_5</math> to a quaternion variable <math>q = x + yr</math> where <math>r^*)2 = -1</math>. Indeed,and by<math>r hypothesis\in \mathbb{H}</math>.

Then, let <math>r^*</math> represent the conjugate of <math>r</math>, so that <math>q = x - yr^*</math>. The extension to <math>\mathbb{H}</math> will be complete when it is shown that <math>f_5(q) = f_5(x - yr^*)</math>. Indeed, by hypothesis

:<math>u(x,y) = u(x,-y), \quad v(x,y) = -v(x,-y) \quad</math> so that one obtains

:<math>u(x,y) = u(x,-y), \quad v(x,y) = -v(x,-y) \quad</math> one obtains

:<math>f_5(x - y r^*) = u(x,-y) + r^* v(x,-y) = u(x,y) + r v(x,y) = f_5(q).</math>

==Homographies==

In the following, colons and square brackets are used to denote [[homogeneous coordinates|homogeneous vectors]].

The [[quaternions and spatial rotation|rotation]] about axis ''r'' is a classical application of quaternions to [[space]] mapping.<ref>[[Arthur Cayley]] (1848) "On the application of quaternions to the theory of rotation", [[London and Edinburgh Philosophical Magazine]], especially page 198,

[http://www.books.google.com/books?id=kolJAAAAYAAJ Google books link] {{webarchive |url=https://web.archive.org/web/20140617191332/http://www.books.google.com/books?id=kolJAAAAYAAJ |date=June 17, 2014 }}</ref> In terms of a [[Homography#Over a ring|homography]], the rotation is expressed

The [[quaternions and spatial rotation|rotation]] about axis ''r'' is a classical application of quaternions to [[space]] mapping.<ref>{{harv|Cayley|1848|loc=especially page 198}}</ref>

:<math>U(q,1) \begin{pmatrix}u & 0\\0 & u \end{pmatrix} = U(qu,u) \thicksim U(u^{-1}qu, 1) ,</math>

In terms of a [[Homography#Over a ring|homography]], the rotation is expressed

:<math>[q:1] \begin{pmatrix}u & 0\\0 & u \end{pmatrix} = [qu : u] \thicksim [u^{-1}qu : 1] ,</math>

where <math>u = \exp(\theta r) = \cos \theta + r \sin \theta</math> is a [[versor]]. If ''p'' * = &minus;''p'', then the translation <math>q \mapsto q + p</math> is expressed by

:<math>U([q, : 1)]\begin{pmatrix}1 & 0 \\ p & 1 \end{pmatrix} = U([q + p, : 1)].</math>

Rotation and translation ''xr'' along the axis of rotation is given by

:<math>U([q, : 1)]\begin{pmatrix}u & 0 \\ uxr & u \end{pmatrix} = U([qu + uxr, : u)] \thicksim U([u^{-1}qu + xr, : 1)].</math>

Such a mapping is called a [[screw displacement]]. In classical [[kinematics]], [[Chasles' theorem (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<ref>{{harv|Hamilton |1853 |loc=§287 pagespp. 273,4}}</ref> by the homography composition

:<math>\begin{pmatrix}1 & 0 \\ -s & 1 \end{pmatrix} \begin{pmatrix}u & 0 \\ 0 & u \end{pmatrix} \begin{pmatrix}1 & 0 \\ s & 1 \end{pmatrix} = \begin{pmatrix}u & 0 \\ z & u \end{pmatrix}, </math>

where <math>z = u s - s u = \sin \theta (rs - sr) = 2 t \sin \theta .</math>

Line 59 ⟶ 58:

Any ''p'' in this half-plane lies on a ray from the origin through the circle <math>\lbrace u^{-1} z : 0 < \theta < \pi \rbrace</math> and can be written <math>p = a u^{-1} z , \ \ a > 0 .</math>

Then ''up'' = ''az'', with <math>\begin{pmatrix}u & 0 \\ az & u \end{pmatrix} </math> as the homography expressing [[conjugation (group theory)|conjugation]] of a rotation by a translation p.

== Linear map ==

The map <math>f:\mathbb H\rightarrow \mathbb H</math>

of quaternion algebra is called linear, if following equalities hold

: <math>f(x+y)=f(x)+f(y)</math>

: <math>f(ax)=af(x)</math>

: <math>x,y\in\mathbb H, a\in\mathbb R</math>

where <math>\mathbb R</math> is real field.

Since <math>f</math> is linear map of quaternion algebra,

then, for any <math>a, b\in\mathbb H</math>, the map

: <math>(afb)(x)=af(x)b</math>

is linear map.

If <math>f</math> is identity map (<math>f(x)=x</math>),

then, for any <math>a, b\in\mathbb H</math>,

we identify tensor product <math>a\otimes b</math> and the map

: <math>(a\otimes b)\circ x=axb</math>

For any linear map

<math>f:\mathbb H\rightarrow \mathbb H</math>

there exists a tensor <math>a\in\mathbb H\otimes\mathbb H</math>,

<math>a=\sum_s a_{s0}\otimes a_{s1}</math>,

such that

: <math>f(x)=a\circ x=(\sum_s a_{s0}\otimes a_{s1})\circ x=\sum_s a_{s0}xa_{s1}</math>

So we can identify the linear map <math>f</math>

and the tensor <math>a</math>.

== 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 <math>\ f(q) = q^2\ </math> from differentiabilitydifferentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.<ref>W.R. {{harvp|Hamilton (1899) ''Elements of Quaternions'' v. I, edited by Charles Jasper Joly|1866|loc=Chapter&nbsp;II, "On differentials and developments of functions of quaternions", pagespp.&nbsp;391–495 430–64}}</ref><ref>[[Charles-Ange {{harvp|Laisant]] (|1881) [https://books.google.com/books?id|loc=WvMGAAAAYAAJ Introduction a la Méthode des Quaternions], Chapitre &nbsp;5: Différentiation des Quaternions, pp.&nbsp;104–117 104–17, link from [[Google Books]]}}</ref>

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:

Considering of the increment of polynomial function of quaternionic argument shows

that the increment is linear map of increment of the argument.

This statement is the basis for the following definition.

A continuous function

Continuous map

<math>\ f: \mathbb H \rightarrow \mathbb H\ </math>

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

is called differentiable

: <math> f(x+h) - f(x) = \frac{\operatorname d f(x)}{\operatorname d x} \circ h + o(h)</math>

on the set <math>U\subset \mathbb H</math>,

if, at every point <math>x\in U</math>,

the increment of the map <math>f</math> can be represented as

: <math>f(x+h)-f(x)=\frac{d f(x)}{d x}\circ h+o(h)</math>

where

: <math>\frac{\operatorname d f(x)}{\operatorname d x}:\mathbb H\rightarrow\mathbb H</math>

is [[linear map]] of quaternion algebra <math>\ \mathbb H\ ,</math> and

<math>\ o:\mathbb H\rightarrow \mathbb H\ </math>

isrepresents suchsome continuous map such that

: <math>\lim_{a \rightarrow 0} \frac{\ \left|\ o(a)\ \right|\ }{ \left|\ a\ \right| } = 0\ ,</math>

and the notation <math>\ \circ h\ </math> denotes ...{{explain|date=September 2024}}

Linear map

<math>\frac{d f(x)}{d x}</math>

The linear map

is called derivative of the map <math>f</math>.

<math>\frac{\operatorname d f(x)}{\operatorname d x}</math>

is called the derivative of the map <math>\ f ~.</math>

On the quaternions, the derivative may be expressed as

: <math>\frac{\operatorname d f(x)}{\operatorname d x} = \sum_s \frac{\operatorname{d}_{s0} f(x)}{\operatorname d x} \otimes \frac{\operatorname{d}_{s1} f(x)}{\operatorname d x} </math>

: <math>\frac{d f(x)}{d x}=

Therefore, the differential of the map <math>\ f\ </math> may be expressed as follows, with brackets on either side.

\sum_s \frac{d_{s0} f(x)}{d x}

:<math>\frac{\operatorname d f(x)}{\operatorname d x}\circ \operatorname d x = \left(\sum_s \frac{\operatorname{d}_{s0} f(x)}{\operatorname d x} \otimes \frac{\operatorname{d}_{s1} f(x)}{\operatorname d x}\right)\circ \operatorname d x = \sum_s \frac{\operatorname{d}_{s0} f(x)}{\operatorname d x} \left( \operatorname d x \right) \frac{\operatorname{d}_{s1} f(x)}{\operatorname d x}</math>

\otimes

\frac{d_{s1} f(x)}{d x}

</math>

Therefore, the differential of the map <math>f</math> may be expressed as

</math>

:<math>\frac{d f(x)}{d x}\circ dx=

\left(

\sum_s \frac{d_{s0} f(x)}{d x}

\otimes

\frac{d_{s1} f(x)}{d x}\right)\circ dx=

\sum_s \frac{d_{s0} f(x)}{d x}

dx

\frac{d_{s1} f(x)}{d x}

</math>

The number of terms in the sum will depend on the function ''<math>\ f'' ~.</math> The expressions

<math>~~ \frac{d_\operatorname{d}_{sp}\operatorname d f(x)}{\operatorname d x}, ~~ \mathsf{\ for\ } ~~ p = 0, 1 ~~</math> are called

components of derivative.

The derivative of a quaternionic function holdsis thedefined followingby the equalitiesexpression

: <math>\frac{df\operatorname d f(x)}{\operatorname d x}\circ h = \lim_{t\to 0}\left(t^\ \frac{-1}(\ f(x +th t\ h) - f(x))\ }{ t }\ \right)</math>

where the variable <math>\ t\ </math> is a real scalar.

The following equations then hold:

: <math>\frac{d(f(x)+g(x))}{d x}

: <math>\frac{\operatorname d\left( f(x) + g(x) \right)}{\operatorname d x} = \frac{df\operatorname d f(x)}{\operatorname d x} + \frac{dg\operatorname d g(x)}{\operatorname d x}</math>

: <math>\frac{\operatorname d\left( f(x)\ g(x)\right)}{\operatorname d x} = \frac{\operatorname d f(x) }{\operatorname d x}\ g(x) + f(x)\ \frac{\operatorname d g(x)}{\operatorname d x}</math>

: <math>\frac{df(x)g(x)}{d x}

=\frac{df(x)}{d x}\ g(x)+f(x)\ \frac{dg(x)}{d x}</math>

: <math>\frac{\operatorname d \left( f(x)\ g(x)\right)}{\operatorname d x} \circ h = \left(\frac{\operatorname d f(x)}{\operatorname d x}\circ h\right )\ g(x) + f(x) \left(\frac{\operatorname d g(x)}{\operatorname d x}\circ h\right)</math>

: <math>\frac{df(x)g(x)}{d x} \circ h

=\left(\frac{df(x)}{d x}\circ h\right )\ g(x)+f(x)\left(\frac{dg(x)}{d x}\circ h\right)</math>

: <math>\frac{\operatorname d \left( a\ f(x)\ b\right)}{\operatorname d x} = a\ \frac{\operatorname d f(x)}{\operatorname d x}\ b</math>

: <math>\frac{daf(x)b}{d x}

=a\ \frac{df(x)}{d x}\ b</math>

: <math>\frac{\operatorname d \left( a\ f(x)\ b\right)}{\operatorname d x}\circ h = a \left(\frac{\operatorname d f(x)}{\operatorname d x}\circ h\right) b</math>

: <math>\frac{daf(x)b}{d x}\circ h

=a\left(\frac{df(x)}{d x}\circ h\right) b</math>

For the function ''<math>\ f''(''x'') = ''axb''a\ x\ b\ ,</math> where <math>\ a\ </math> and <math>\ b\ </math> are constant quaternions, the derivative is

{| class="wikitable"

|-

| <math> \frac{daxb\operatorname d \left( a\ x\ b \right)}{\operatorname d x} = a \otimes b </math>

|style="background:white;"| &emsp;

|

| <math>dy \operatorname d y=\frac{daxb\operatorname d \left(a\ x\ b\right)}{\operatorname d x} \circ dx\operatorname d x = a\,dx \,left(\operatorname d x\right)\ b</math>

|}

Line 160 ⟶ 115:

{| class="wikitable"

|-

| <math> \frac{d_\operatorname{d}_{10} axb\left(a\ x\ b\right)}{\operatorname d x} = a </math>

|style="background:white;"| &emsp;

|

| <math> \frac{d_\operatorname{d}_{11} axb\left(a\ x\ b\right)}{\operatorname d x} = b </math>

|}

Similarly, for the function ''<math>\ f''(''x'') = ''x^2''<sup>2\ ,</supmath>, the derivative is

{| class="wikitable"

|-

| <math>\frac{dx\operatorname d x^2}{\operatorname d x}=x \otimes 1 + 1 \otimes x</math>

|style="background:white;"| &emsp;

|

| <math>dy\operatorname d y=\frac{dx\operatorname d x^2}{\operatorname d x}\circ dx\operatorname d x = x\,dx \operatorname d x +dx (\,operatorname d x)\ x </math>

|}

Line 176 ⟶ 131:

{| class="wikitable"

|-

| <math> \frac{d_\operatorname{d}_{10} x^2 }{\operatorname d x} = x </math>

|style="background:white;"| &emsp;

|

| <math> \frac{d_\operatorname{d}_{11} x^2}{\operatorname d x} = 1 </math>

|-

| <math>\frac{d_\operatorname{d}_{20} x^2 }{\operatorname d x} = 1 </math>

|style="background:white;"| &emsp;

|

| <math>\frac{d_\operatorname{d}_{21} x^2}{\operatorname d x} = x </math>

|}

Finally, for the function '' <math>\ f''(''x'') = ''x''<sup>&minus;^{-1}\ ,</supmath>, the derivative is

{| class="wikitable"

|-

| <math> \frac{dx\operatorname d x^{-1} }{\operatorname d x} = -x^{-1} \otimes x^{-1}</math>

|style="background:white;"| &emsp;

|

| <math>dy \operatorname d y = \frac{dx\operatorname d x^{-1} }{\operatorname d x} \circ dx\operatorname d x = -x^{-1}dx(\,operatorname d x)\ x^{-1}</math>

|}

Line 196 ⟶ 151:

{| class="wikitable"

|-

| <math>\frac{d_\operatorname{d}_{10} x^{-1} }{\operatorname d x} = -x^{-1} </math>

|style="background:white;"| &emsp;

|

| <math>\frac{d_\operatorname{d}_{11} x^{-1} }{\operatorname d x} = x^{-1} </math>

|}

==See also==

* [[Cayley transform]]

* [[Quaternionic manifold]]

==Notes==

{{notelist}}

==Citations==

{{Reflist|2}}

==References==

{{Reflist}}

* {{Citation

* [[Vladimir Arnold]] (1995) "The geometry of spherical curves and the algebra of quaternions", translated by [[Ian R. Porteous]], ''Russian Mathematical Surveys'' 50:1&ndash;68.

| last = Arnold

* Graziano Gentili, Catarina Stoppato & D.C. Struppa (2013) ''Regular Functions of a Quaternionic Variable'', Birkhäuser, {{isbn|978-3-642-33870-0}}.

| first = Vladimir

* P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", ''Proceedings of the [[Royal Irish Academy]]'', Section A 51: 67&ndash;85.

| author-link = Vladimir Arnold

* K. Gürlebeck & W. Sprössig (1990) ''Quaternionic analysis and elliptic boundary value problems'', Birkhäuser {{isbn|978-3-7643-2382-0}}.

| translator1-last = Porteous

* [[W. R. Hamilton]] (1853) [https://web.archive.org/web/20140808040037/http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2 Lectures on Quaternions], Royal Irish Academy, weblink from [[Cornell University]] ''Historical Math Monographs''.

| translator1-first = Ian R.

* [[Charles Jasper Joly]] (1903) [https://www.jstor.org/stable/90902 Quaternions and Projective Geometry], [[Philosophical Transactions of the Royal Society of London]] 201:223–327.

| translator-link1 = Ian R. Porteous

* R. Michael Porter (1998) [http://www.ams.org/journals/ecgd/1998-02-06/S1088-4173-98-00032-0/S1088-4173-98-00032-0.pdf Möbius invariant quaternion geometry], ''Conformal Geometry and Dynamics'' 2:89&ndash;196.

| title = The geometry of spherical curves and the algebra of quaternions

* A. Sudbery (1979) "Quaternionic Analysis", ''Mathematical Proceedings of the Cambridge Philosophical Society'' 85:199&ndash;225.

| journal = Russian Mathematical Surveys

* [[Patrick du Val]] (1964) ''Homographies, quaternions and rotations'', Oxford Mathematical Monographs, [[Oxford University Press|Clarendon Press]], [[Oxford]], {{MathSciNet|id=0169108}} .

| volume = 50

| issue = 1

| pages = 1&ndash;68

| year = 1995

| doi = 10.1070/RM1995v050n01ABEH001662

| zbl = 0848.58005| s2cid = 250897899

}}

* {{Citation

| last = Cayley

| first = Arthur

| author-link = Arthur Cayley

| title = On the application of quaternions to the theory of rotation

| journal = [[London and Edinburgh Philosophical Magazine]] |series=Series 3

| volume = 33

| issue = 221

| pages = 196&ndash;200

| year = 1848

| doi = 10.1080/14786444808645844| url = https://zenodo.org/record/1782007

}}

*{{Citation

| last = Deavours

| first = C.A.

| title = The quaternion calculus

| journal = [[American Mathematical Monthly]]

| publisher = Mathematical Association of America

| ___location = Washington, DC

| issn = 0002-9890

| volume = 80

| issue = 9

| pages = 995–1008

| year = 1973

| jstor = 2318774

| doi = 10.2307/2318774

| zbl = 0282.30040}}

* {{Citation

| last = Du Val

| first = Patrick

| author-link = Patrick du Val

| title = Homographies, Quaternions and Rotations

| place = Oxford

| publisher = Clarendon Press

| series = Oxford Mathematical Monographs

| year = 1964

| mr = 0169108

| zbl = 0128.15403}}

* {{Citation

| last = Fueter

| first = Rudolf

| author-link = Rudolf Fueter

| title = Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen

| journal = [[Commentarii Mathematici Helvetici]]

| volume = 8

| pages = 371–378

| year = 1936

| language = de

| zbl = 0014.16702| doi = 10.1007/BF01199562

| s2cid = 121227604

}}

* {{Citation

| last1 = Gentili

| first1 = Graziano

| last2 = Stoppato

| first2 = Caterina

| last3 = Struppa

| first3 = Daniele C.

| title = Regular Functions of a Quaternionic Variable

| place = Berlin

| publisher = Springer

| year = 2013

| doi = 10.1007/978-3-642-33871-7

| isbn = 978-3-642-33870-0

| zbl = 1269.30001| s2cid = 118710284

}}

* {{Citation

| last = Gormley

| first = P.G.

| title = Stereographic projection and the linear fractional group of transformations of quaternions

| journal = [[Proceedings of the Royal Irish Academy, Section A]]

| volume = 51

| pages = 67&ndash;85

| year = 1947

| jstor = 20488472

}}

* {{Citation

| last1 = Gürlebeck

| first1 = Klaus

| last2 = Sprößig

| first2 = Wolfgang

| title = Quaternionic analysis and elliptic boundary value problems

| place = Basel

| publisher = Birkhäuser

| year = 1990

| isbn = 978-3-7643-2382-0

| zbl = 0850.35001}}

* {{citation | last =John C.Holladay| title =The Stone–Weierstrass theorem for quaternions| journal = Proc. Amer. Math. Soc. | volume =8| year =1957|url =http://www.ams.org/journals/proc/1957-008-04/S0002-9939-1957-0087047-7/S0002-9939-1957-0087047-7.pdf| doi=10.1090/S0002-9939-1957-0087047-7| pages=656| doi-access =free}}.

* {{Citation

| last = Hamilton

| first = William Rowan

| author-link = William Rowan Hamilton

| title = Lectures on Quaternions

| place = Dublin

| publisher = Hodges and Smith

| year = 1853

| ol = 23416635M

}}

* {{Citation

| last = Hamilton

| first = William Rowan

| author-link = William Rowan Hamilton

| editor-last = Hamilton

| editor-first = William Edwin

| editor-link = William Edwin Hamilton

| title = Elements of Quaternions

| place = London

| publisher = Longmans, Green, & Company

| year = 1866

| url = https://books.google.com/books?id=b2stAAAAYAAJ

| zbl = 1204.01046}}

* {{Citation

| last = Joly

| first = Charles Jasper

| author-link = Charles Jasper Joly

| title = Quaternions and projective geometry

| journal = [[Philosophical Transactions of the Royal Society of London]]

| volume = 201

| issue = 331–345

| pages = 223–327

| year = 1903

| jstor = 90902

| doi = 10.1098/rsta.1903.0018

| jfm = 34.0092.01| bibcode = 1903RSPTA.201..223J

| doi-access =

}}

* {{Citation

| last = Laisant

| first = Charles-Ange

| author-link = Charles-Ange Laisant

| title = Introduction à la Méthode des Quaternions

| place = Paris

| publisher = Gauthier-Villars

| year = 1881

| language = fr

| url = https://archive.org/details/introductionlam01laisgoog

| jfm = 13.0524.02}}

* {{Citation

| last = Porter

| first = R. Michael

| title = Möbius invariant quaternion geometry

| journal = Conformal Geometry and Dynamics

| volume = 2

| issue = 6

| pages = 89&ndash;196

| year = 1998

| url = https://www.ams.org/journals/ecgd/1998-02-06/S1088-4173-98-00032-0/S1088-4173-98-00032-0.pdf

| doi = 10.1090/S1088-4173-98-00032-0

| zbl = 0910.53005| doi-access = free

}}

* {{Citation

| last = Sudbery

| first = A.

| title = Quaternionic analysis

| journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]

| volume = 85

| issue = 2

| pages = 199&ndash;225

| year = 1979

| doi = 10.1017/S0305004100055638

| zbl = 0399.30038| bibcode = 1979MPCPS..85..199S

| hdl = 10338.dmlcz/101933

| s2cid = 7606387

| hdl-access = free

}}

{{Analysis in topological vector spaces}}

==Notes==

{{notelist}}

[[Category:Quaternions]]

[[Category:Functions and mappings]]

[[Category:Articles containing proofs]]

[[Category:Functions and mappings]]

[[Category:Quaternions|analysis]]