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{{Short description|Study of analytic functions of quaternions, generalizing complex analysis}}
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
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
==Properties==
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f_4(j) = -j, \quad
f_4(k) = -k </math>.
Consequently, since <math>f_4</math> is [[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>
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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>{{harv|Hamilton|1866|loc=Chapter II, On differentials and developments of functions of quaternions, pp. 391–495}}</ref><ref>{{harv|Laisant|1881|loc=Chapitre 5: Différentiation des Quaternions, pp. 104–117}}</ref>
Considering of the increment of [[polynomial function]] of quaternionic argument shows that the increment is linear map of increment of the argument. {{Dubious|date=March 2019}} From this, a definition can be made:
<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{d f(x)}{d x}\circ h+o(h)</math>
where
: <math>\frac{d f(x)}{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 such continuous map that
: <math>\lim_{a\rightarrow 0}\frac{|o(a)|}{|a|}=0</math>
<math>\frac{d f(x)}{d x}</math>
is called the derivative of the map <math>f</math>.
On the quaternions, the derivative may be expressed as
: <math>\frac{d f(x)}{d x} = \sum_s \frac{d_{s0} f(x)}{d x} \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>
▲Therefore, the differential of the map <math>f</math> may be expressed as
The number of terms in the sum will depend on the function ''f''. The expressions
<math>\frac{d_{sp}d f(x)}{d x}, p = 0,1</math> are called
components of derivative.
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: <math>\frac{df(x)}{d x}\circ h=\lim_{t\to 0}(t^{-1}(f(x+th)-f(x)))</math>
: <math>\frac{d(f(x)+g(x))}{d x} = \frac{df(x)}{d x}+\frac{dg(x)}{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{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{daf(x)b}{d x} = a\ \frac{df(x)}{d x}\ 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 ''f''(''x'') = ''axb'', the derivative is
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Finally, for the function ''f''(''x'') = ''x''<sup>
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