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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Motor variable ： ウィキペディア英語版 Motor variable In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. ''Motor variable'' is used here in place of ''split-complex variable'' for euphony and tradition. For example, :$f(z)\; =\; u(z)\; +\; j\; \backslash \; v(z)\; ,\backslash \; z\; =\; x\; +\; j\; y\; ,\backslash \; x,y\; \backslash in\; R\; ,\backslash quad\; j^2\; =\; +1,\backslash quad\; u(z),v(z)\; \backslash in\; R.$ Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with motor variables. ==Elementary functions of a motor variable== Let D = $\backslash$, the split-complex plane. The following exemplar functions ''f'' have domain and range in D: The action of a hyperbolic versor $u\; =\; \backslash exp(aj)\; =\; \backslash cosh\; a\; +\; j\; \backslash sinh\; a$ is combined with translation to produce the affine transformation :$f(z)\; =\; uz\; +\; c\; \backslash$. When ''c'' = 0, the function is equivalent to a squeeze mapping. The squaring function has no analogy in ordinary complex arithmetic. Let :$f(z)\; =\; z^2\; \backslash$ and note that $f(-1)=f(j)=\; f(-j)\; =\; 1.\; \backslash$ The result is that the four quadrants are mapped into one, the identity component: :$U\_1\; =\; \backslash$. Note that $z\; z^$ * = 1 \ forms the unit hyperbola $x^2\; -\; y^2\; =\; 1$. Thus the reciprocation :$f(z)\; =\; 1/z\; =\; z^$ */\mid z \mid^2 \text \mid z \mid^2 = z z^ * involves the hyperbola as curve of reference as opposed to the circle in C. On the extended complex plane one has the class of functions called Möbius transformations: :$f(z)\; =\; \backslash frac\; .$ Using the concept of a projective line over a ring, the projective line P(D) is formed and acted on by the group of homographies GL(2,D). The construction uses homogeneous coordinates with split-complex number components. On the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it. A mapping of the identity component U1 into a rectangle provides a comparable bounding action: :$f(z)\; =\; \backslash frac\; ,\; \backslash quad\; f:U\_1\; \backslash to\; T$ where T = . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Motor variable」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.