Quaternionic analysis: Difference between revisions

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{{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 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. Unlike the [[complex numbersnumber]]s and like the [[real number|reals]], the four notions do not coincide.

==Properties==

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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 as with other number systems, <math>f_2(0)</math> and related problems are generally discludedexcluded due to the nature of [[dividing by zero]].

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

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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>

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 asAs a union of complex planes|appears as a union of complex planes]], the following proposition shows that extending complex functions requires special care:

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>.

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==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>{{harv|Cayley|1848|loc=especially page 198}}</ref>

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

:<math>U[q,\ :1] \begin{pmatrix}u & 0\\0 & u \end{pmatrix} = U[qu,\ : u] \thicksim U[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''.

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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 differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.<ref>{{harvharvp|Hamilton|1866|loc=Chapter &nbsp;II, On differentials and developments of functions of quaternions, pp. &nbsp;391–495 }}</ref><ref>{{harvharvp|Laisant|1881|loc=Chapitre &nbsp;5: Différentiation des Quaternions, pp. &nbsp;104–117 }}</ref>

Considering of 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

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 With Brackets on either side.

:<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>

|}

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{| 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<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>

|}

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{| 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>

|}

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{| 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==

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| pages = 1&ndash;68

| year = 1995

| doi = 10.1070/RM1995v050n01ABEH001662 | id =

| zbl = 0848.58005| s2cid = 250897899

| mr =

}}

| zbl = 0848.58005}}

* {{Citation

| last = Cayley

Line 247 ⟶ 210:

| jstor = 2318774

| doi = 10.2307/2318774

| mr =

| zbl = 0282.30040}}

* {{Citation

Line 269 ⟶ 231:

| pages = 371–378

| year = 1936

| language = Germande

| mr =

| zbl = 0014.16702| doi = 10.1007/BF01199562

| s2cid = 121227604

}}

* {{Citation

| lastlast1 = Gentili

| firstfirst1 = Graziano

| last2 = Stoppato

| first2 = CatarinaCaterina

| last3 = Struppa

| first3 = Daniele C.

Line 286 ⟶ 248:

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

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

| zbl = 1269.30001| s2cid = 118710284

| mr =

}}

| zbl = 1269.30001}}

* {{Citation

| last = Gormley

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| year = 1947

| jstor = 20488472

|}} mr =

| zbl = }}

* {{Citation

| lastlast1 = Gürlebeck

| firstfirst1 = Klaus

| last2 = Sprößig

| first2 = Wolfgang

Line 309 ⟶ 270:

| year = 1990

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

| mr =

| 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

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| publisher = Hodges and Smith

| year = 1853

| ol = 23416635M

| url = https://openlibrary.org/books/OL23416635M/Lectures_on_quaternions}}

}}

* {{Citation

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* {{Citation

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| year = 1881

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

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| pages = 89&ndash;196

| year = 1998

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

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* {{Citation

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

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| s2cid = 7606387

| hdl-access = free

}}

{{Analysis in topological vector spaces}}

[[Category:Quaternions]]

[[Category:Functions and mappings]]

[[Category:Articles containing proofs]]

[[Category:Functions and mappings]]

[[Category:Quaternions|analysis]]