Quaternion Analysis

Quaternion analysis is a branch of mathematics that studies the regular quaternion- valued functions of a quaternion variable. Due to the noncommutativity of quaternion algebra, there are various non-equivalent approaches to the definition of regular quaternion functions. This article will mainly deal with Fueter's approach ^[1] .

Defining a regular function

Consider the operator

{\bar {\partial }}={\frac {\partial }{\partial {\bar {q}}}}={\frac {\partial }{\partial t}}+{\vec {i}}{\frac {\partial }{\partial x}}+{\vec {j}}{\frac {\partial }{\partial y}}+{\vec {k}}{\frac {\partial }{\partial z}}

{\ displaystyle {\ bar {\ partial}} = {\ frac {\ partial} {\ partial {\ bar {q}}}} = {\ frac {\ partial} {\ partial t}} + {\ vec { i}} {\ frac {\ partial} {\ partial x}} + {\ vec {j}} {\ frac {\ partial} {\ partial y}} + {\ vec {k}} {\ frac {\ partial} {\ partial z}}}

{\bar \partial }={\frac {\partial }{\partial {\bar q}}}={\frac {\partial }{\partial t}}+{\vec i}{\frac {\partial }{\partial x}}+{\vec j}{\frac {\partial }{\partial y}}+{\vec k}{\frac {\partial }{\partial z}}

Quaternion Variable Function $f\colon \mathbb {H} \to \mathbb {H}$ ${\ displaystyle f \ colon \ mathbb {H} \ to \ mathbb {H}}$ $f\colon \mathbb {H} \to \mathbb {H}$ called regular if

{\bar {\partial }}f=0

{\ displaystyle {\ bar {\ partial}} f = 0}

{\bar \partial }f=0

Harmonic functions

Let be ${\bar {\partial }}f=0$ ${\ displaystyle {\ bar {\ partial}} f = 0}$ ${\bar \partial }f=0$ then $\partial {\bar {\partial }}f=0$ ${\ displaystyle \ partial {\ bar {\ partial}} f = 0}$ $\partial {\bar \partial }f=0$ . It is easy to verify that the operator $\partial {\bar {\partial }}$ ${\ displaystyle \ partial {\ bar {\ partial}}}$ $\partial {\bar \partial }$ has the form

\partial {\bar {\partial }}={\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}=\Delta _{4}

{\ displaystyle \ partial {\ bar {\ partial}} = {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial x ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial y ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial z ^ {2}} } = \ Delta _ {4}}

\partial {\bar \partial }={\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}=\Delta _{4}

and coincides with the Laplace operator in $\mathbb {R} ^{4}$ ${\ displaystyle \ mathbb {R} ^ {4}}$ $\mathbb {R} ^{4}$ . Thus, all components of a regular quaternion function are harmonic functions in $\mathbb {R} ^{4}$ ${\ displaystyle \ mathbb {R} ^ {4}}$ $\mathbb {R} ^{4}$ . Conversely, it can be shown that for any harmonic function $\tau \colon \mathbb {R} ^{4}\to \mathbb {R}$ ${\ displaystyle \ tau \ colon \ mathbb {R} ^ {4} \ to \ mathbb {R}}$ $\tau \colon \mathbb {R} ^{4}\to \mathbb {R}$ there is a regular quaternion function $f$ ${\ displaystyle f}$ $f$ such that $\tau =\operatorname {Scal} \,f$ ${\ displaystyle \ tau = \ operatorname {Scal} \, f}$ $\tau =\operatorname {Scal}\,f$ . Many properties of regular quaternion functions, in particular, the maximum principle , immediately follow from the properties of harmonic functions.

Some applications

Quaternions are actively used to calculate three-dimensional graphics in computer games.

Differentiating mappings

Let be $y=f(x)$ ${\ displaystyle y = f (x)}$ Is a function defined on the body of quaternions. We can define the concept of the left derivative $y'_{l}$ ${\ displaystyle y '_ {l}}$ at the point $x=a$ ${\ displaystyle x = a}$ like the number that

f(x)-f(a)=y'_{l}(x-a)+o(x-a)

{\ displaystyle f (x) -f (a) = y '_ {l} (xa) + o (xa)}

Where $o(h)$ ${\ displaystyle o (h)}$ - infinitesimal from $h$ ${\ displaystyle h}$ , i.e

\lim _{h\to 0}{\frac {|o(h)|}{|h|}}=0

{\ displaystyle \ lim _ {h \ to 0} {\ frac {| o (h) |} {| h |}} = 0}

.

Many functions that have a left derivative are bounded. For example, features like

y=axb

{\ displaystyle y = axb}

y=x^{2}

{\ displaystyle y = x ^ {2}}

do not have a left derivative.

Consider the increment of these functions more closely.

a(x+h)b-axb=ahb

{\ displaystyle a (x + h) b-axb = ahb}

(x+h)^{2}-x^{2}=xh+hx+h^{2}

{\ displaystyle (x + h) ^ {2} -x ^ {2} = xh + hx + h ^ {2}}

It is easy to verify that the expressions

a h b

{\ displaystyle ahb}

and

xh+hx

{\ displaystyle xh + hx}

are linear functions of a quaternion $h$ ${\ displaystyle h}$ . This observation is the basis for the following definition ^[2] .

Continuous display

f:\mathbb {H} \rightarrow \mathbb {H}

{\ displaystyle f: \ mathbb {H} \ rightarrow \ mathbb {H}}

called differentiable on the set $U\subset \mathbb {H}$ ${\ displaystyle U \ subset \ mathbb {H}}$ if at every point $x\in U$ ${\ displaystyle x \ in U}$ display change $f$ ${\ displaystyle f}$ can be represented as

f(x+h)-f(x)={\frac {df(x)}{dx}}\circ h+o(h)

{\ displaystyle f (x + h) -f (x) = {\ frac {df (x)} {dx}} \ circ h + o (h)}

Where

{\frac {df(x)}{dx}}:\mathbb {H} \rightarrow \mathbb {H}

{\ displaystyle {\ frac {df (x)} {dx}}: \ mathbb {H} \ rightarrow \ mathbb {H}}

linear mapping of quaternion algebra $\mathbb {H}$ ${\ displaystyle \ mathbb {H}}$ and $o:\mathbb {H} \rightarrow \mathbb {H}$ ${\ displaystyle o: \ mathbb {H} \ rightarrow \ mathbb {H}}$ such a continuous mapping that

\lim _{a\rightarrow 0}{\frac {|o(a)|}{|a|}}=0

{\ displaystyle \ lim _ {a \ rightarrow 0} {\ frac {| o (a) |} {| a |}} = 0}

Linear mapping

{\frac {df(x)}{dx}}

{\ displaystyle {\ frac {df (x)} {dx}}}

called derivative mapping $f$ ${\ displaystyle f}$ .

The derivative can be represented as ^[3]

{\frac {df(x)}{dx}}={\frac {d_{s0}f(x)}{dx}}\otimes {\frac {d_{s1}f(x)}{dx}}

{\ displaystyle {\ frac {df (x)} {dx}} = {\ frac {d_ {s0} f (x)} {dx}} \ otimes {\ frac {d_ {s1} f (x)} { dx}}}

Respectively display differential $f$ ${\ displaystyle f}$ has the form

df={\frac {df(x)}{dx}}\circ dx=\left({\frac {d_{s0}f(x)}{dx}}\otimes {\frac {d_{s1}f(x)}{dx}}\right)\circ dx={\frac {d_{s0}f(x)}{dx}}dx{\frac {d_{s1}f(x)}{dx}}

{\ displaystyle df = {\ frac {df (x)} {dx}} \ circ dx = \ left ({\ frac {d_ {s0} f (x)} {dx}} \ otimes {\ frac {d_ { s1} f (x)} {dx}} \ right) \ circ dx = {\ frac {d_ {s0} f (x)} {dx}} dx {\ frac {d_ {s1} f (x)} { dx}}}

Index summation is assumed here. $s$ ${\ displaystyle s}$ . The number of terms depends on the choice of function $f$ ${\ displaystyle f}$ . Expressions

{\frac {d_{s0}df(x)}{dx}},{\frac {d_{s1}f(x)}{dx}}

{\ displaystyle {\ frac {d_ {s0} df (x)} {dx}}, {\ frac {d_ {s1} f (x)} {dx}}}

are called components of the derivative.

Derivative satisfies equalities

{\frac {d(f(x)+g(x))}{dx}}={\frac {df(x)}{dx}}+{\frac {dg(x)}{dx}}

{\ displaystyle {\ frac {d (f (x) + g (x))} {dx}} = {\ frac {df (x)} {dx}} + {\ frac {dg (x)} {dx }}}

{\frac {df(x)g(x)}{dx}}={\frac {df(x)}{dx}}\ g(x)+f(x)\ {\frac {dg(x)}{dx}}

{\ displaystyle {\ frac {df (x) g (x)} {dx}} = {\ frac {df (x)} {dx}} \ g (x) + f (x) \ {\ frac {dg (x)} {dx}}}

{\frac {df(x)g(x)}{dx}}\circ h=\left({\frac {df(x)}{dx}}\circ h\right)\ g(x)+f(x)\left({\frac {dg(x)}{dx}}\circ h\right)

{\ displaystyle {\ frac {df (x) g (x)} {dx}} \ circ h = \ left ({\ frac {df (x)} {dx}} \ circ h \ right) \ g (x ) + f (x) \ left ({\ frac {dg (x)} {dx}} \ circ h \ right)}

{\frac {daf(x)b}{dx}}=a\ {\frac {df(x)}{dx}}\ b

{\ displaystyle {\ frac {daf (x) b} {dx}} = a \ {\ frac {df (x)} {dx}} \ b}

{\frac {daf(x)b}{dx}}\circ h=a\left({\frac {df(x)}{dx}}\circ h\right)b

{\ displaystyle {\ frac {daf (x) b} {dx}} \ circ h = a \ left ({\ frac {df (x)} {dx}} \ circ h \ right) b}

If a $y=axb$ ${\ displaystyle y = axb}$ , then the derivative has the form

{\frac {daxb}{dx}}=a\otimes b,dy={\frac {daxb}{dx}}\circ dx=a\,dx\,b

{\ displaystyle {\ frac {daxb} {dx}} = a \ otimes b, dy = {\ frac {daxb} {dx}} \ circ dx = a \, dx \, b}

{\frac {d_{10}axb}{dx}}=a,{\frac {d_{11}axb}{dx}}=b

{\ displaystyle {\ frac {d_ {10} axb} {dx}} = a, {\ frac {d_ {11} axb} {dx}} = b}

If a $y=x^{2}$ ${\ displaystyle y = x ^ {2}}$ then the derivative has the form

{\frac {dx^{2}}{dx}}=x\otimes 1+1\otimes x,dy={\frac {dx^{2}}{dx}}\circ dx=x\,dx+dx\,x

{\ displaystyle {\ frac {dx ^ {2}} {dx}} = x \ otimes 1 + 1 \ otimes x, dy = {\ frac {dx ^ {2}} {dx}} \ circ dx = x \ , dx + dx \, x}

and the components of the derivative have the form

{\frac {d_{10}x^{2}}{dx}}=x,{\frac {d_{11}x^{2}}{dx}}=1

{\ displaystyle {\ frac {d_ {10} x ^ {2}} {dx}} = x, {\ frac {d_ {11} x ^ {2}} {dx}} = 1}

{\frac {d_{20}x^{2}}{dx}}=1,{\frac {d_{21}x^{2}}{dx}}=x

{\ displaystyle {\ frac {d_ {20} x ^ {2}} {dx}} = 1, {\ frac {d_ {21} x ^ {2}} {dx}} = x}

If a $y=x^{-1}$ ${\ displaystyle y = x ^ {- 1}}$ , then the derivative has the form

{\frac {dx^{-1}}{dx}}=-x^{-1}\otimes x^{-1},dy={\frac {dx^{-1}}{dx}}\circ dx=-x^{-1}dx\,x^{-1}

{\ displaystyle {\ frac {dx ^ {- 1}} {dx}} = - x ^ {- 1} \ otimes x ^ {- 1}, dy = {\ frac {dx ^ {- 1}} {dx }} \ circ dx = -x ^ {- 1} dx \, x ^ {- 1}}

and the components of the derivative are

{\frac {d_{10}x^{-1}}{dx}}=-x^{-1},{\frac {d_{11}x^{-1}}{dx}}=x^{-1}

{\ displaystyle {\ frac {d_ {10} x ^ {- 1}} {dx}} = - x ^ {- 1}, {\ frac {d_ {11} x ^ {- 1}} {dx}} = x ^ {- 1}}

Notes

↑ Fueter, R. Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen // Commentarii Mathematici Helvetici. - №1. - Birkhäuser Basel, 1936. - V. 8. 8. P. 371-378.
↑ Aleks Kleyn , eprint arXiv: 1601.03259 Introduction into Calculus over Banach algebra, 2016
↑ Expression ${\frac {d_{sp}f(x)}{dx}}$ ${\ displaystyle {\ frac {d_ {sp} f (x)} {dx}}}$ is not a fraction and should be taken as a single symbol. This designation is proposed for compatibility with the designation of the derivative. Value of expression ${\frac {d_{sp}f(x)}{dx}}$ ${\ displaystyle {\ frac {d_ {sp} f (x)} {dx}}}$ for a given $x$ ${\ displaystyle x}$ is a quaternion.

Literature

DB Sweetser , Doing Physics with Quaternions
A. Sudbery , Quaternionic Analysis, Department of Mathematics, University of York, 1977.
V.I. Arnold , Geometry of Spherical Curves and Quaternion Algebra , UMN, 1995, 50: 1 (301), 3-68