Quaternionic analysis: Difference between revisions

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{{Short description|StudyFunction oftheory analyticwith functionsquaternion of quaternions, generalizing complex analysisvariable}}

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 functions of a [[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.

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.

As with 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 numbers and like the 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

Line 27 ⟶ 26:

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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== 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

follows 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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| year = 1995

| doi = 10.1070/RM1995v050n01ABEH001662

| zbl = 0848.58005}}| s2cid = 250897899

}}

* {{Citation

| last = Cayley

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| language = de

| zbl = 0014.16702| doi = 10.1007/BF01199562

| s2cid = 121227604

}}

* {{Citation

| lastlast1 = Gentili

| firstfirst1 = Graziano

| last2 = Stoppato

| first2 = Caterina

Line 263 ⟶ 248:

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

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

| zbl = 1269.30001}}| s2cid = 118710284

}}

* {{Citation

| last = Gormley

Line 275 ⟶ 261:

}}

* {{Citation

| lastlast1 = Gürlebeck

| firstfirst1 = Klaus

| last2 = Sprößig

| first2 = Wolfgang

Line 285 ⟶ 271:

| 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

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| publisher = Longmans, Green, & Company

| year = 1866

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

| zbl = 1204.01046}}

* {{Citation

Line 321 ⟶ 308:

| doi = 10.1098/rsta.1903.0018

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

| doi-access = free

}}

* {{Citation

Line 343 ⟶ 330:

| 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

| 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

Line 358 ⟶ 346:

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

| hdl = 10338.dmlcz/101933

| s2cid = 7606387

| hdl-access = free

}}

{{Analysis in topological vector spaces}}

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