Quaternionic analysis: Difference between revisions

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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 II, On differentials and developments of functions of quaternions, pp. 391–495}}</ref><ref>{{harvharvp|Laisant|1881|loc=Chapitre 5: Différentiation des Quaternions, pp. 104–117}}</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:

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

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

is a continuous map such that

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

and the notation <math>\circ h</math> denotes that the direction-dependent quaternionic derivative is oriented in the direction of the quaternion {{mvar|h}}.

The linear map

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

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

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

The number of terms in the sum will depend on the function ''{{mvar|f''}}. 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 holds the following equalities

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

:: where the variable {{mvar|t}} is real / scalar.

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

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

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

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

For the function {{math|''f''(''x'') {{=}} ''axba x b''}}, 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=\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 111 ⟶ 114:

{| 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''}}, 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=\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 127 ⟶ 130:

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

|}

Line 147 ⟶ 150:

{| 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>

|}