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| 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 [[real variable]] or functions of a [[complex variable]] are called.
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| As with complex and real analysis, it is possible to study the concepts of [[analytic function|analyticity]], [[holomorphy]], [[harmonic function|harmonicity]] and [[conformality]] in the context of quaternions. It is known that for the complex numbers, these four notions coincide; however, for the quaternions, and also the real numbers, not all of the notions are the same.
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| ==Discussion==
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. You can either install Word - Press yourself or use free services offered on the web today. This is a service where people write articles using a specific keyword you have given them. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. They provide many such popular products which you can buy for your baby. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Storing write-ups in advance would have to be neccessary with the auto blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. High Quality Services: These companies help you in creating high quality Word - Press websites. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. Contact Infertility Clinic Providing One stop Fertility Solutions at:. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>Internet is not only the source for information, it is also one of the source for passive income. I'm a large fan of using Word - Press to create pretty much any sort of web page. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. If you loved this short article and you would like to receive much more information about [http://aorta.in/WordpressDropboxBackup178167 wordpress backup plugin] assure visit our web site. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010. |
| The [[projection (linear algebra)|projections]] of a quaternion onto its scalar part or onto its vector part, as well as the modulus and [[versor]] functions, are examples that are basic to understanding quaternion structure. An important example of a function of a quaternion variable is
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| :<math>f(q) = u q u^{-1}</math>
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| which [[quaternions and spatial rotation|rotates the vector part of ''q'']] by twice the angle of ''u''.
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| The quaternion inversion <math>f(q) = q^{-1}</math> is another fundamental function, but it introduces questions ''f''(0) = ? and "Solve ''f''(''q'') = 0." [[Affine transformation]]s
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| of quaternions have the form
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| :<math>f(q) = a q + b, \quad a, b \in \mathbb{H}. </math>
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| Linear fractional transformations of quaternions can be represented by elements of the [[matrix ring]] M<sub>2</sub>(H) operating on the [[projective line over a ring|projective line over H]]. For instance, the mappings <math>q \mapsto u q v,</math> where {{mvar|u}} and {{mvar|v}} are fixed [[versor]]s serve to produce the [[elliptic geometry#elliptic space|motions of elliptic space]].
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| Quaternion variable theory differs in some respects from complex variable theory as in this instance: The [[complex conjugate]] mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic operation. Indeed, conjugation changes the [[orientation (mathematics)|orientation]] of plane figures, something that arithmetic functions do not change. In contrast, the quaternion conjugation can be expressed arithmetically:
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| '''Proposition:''' The function <math>f(q) = - \tfrac 1 2 (q + iqi + jqj + kqk)</math> is equivalent to quaternion conjugation.
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| Proof: For the basis elements we have
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| :<math>f(1) = -\tfrac 1 2 (1-1-1-1) = 1, f(i) = -\tfrac 1 2 (i-i+i+i) = -i, f(j) = -j, f(k) = -k </math>.
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| Consequently, since ''f'' is a [[linear function]],
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| :<math> f(q) \ = f(w + xi + yj + zk) \ = w f(1) + x f(i) + y f(j) + z f(k) \ = w - x i - y j - zk \ = q^*.</math>
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| 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. These efforts were summarized in 1973 by C.A. Deavours. He recalls a 1935 issue of [[Commentarii Mathematici Helvetici]] where an alternative theory of "regular functions" was initiated by [[Rudolf Fueter|R. Fueter]] through the idea of [[Morera's theorem]]: quaternion function F is "left regular at ''q'' " when the integral of F vanishes over any sufficiently small [[hypersurface]] containing ''q''. Then the analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] holds: the only quaternion function regular with bounded norm in E<sup>4</sup> 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.
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| Though '''H''' [[quaternion#H as a union of complex planes|appears as a union of complex planes]], the following proposition shows that extending complex functions requires special care:
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| '''Proposition:''' Let <math>f(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 ''u'' is an [[even function]] of ''y'' and that ''v'' is an [[odd function]] of ''y''. Then <math>f(q) = u(x,y) + r \ v(x,y)</math> is an extension of ''f'' to a quaternion variable <math> q = x + y r, \quad r^2 = -1, \quad r \in \mathbb{H} </math>.
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| Proof: Let ''r*'' be the conjugate of ''r'' so that ''q'' = ''x'' − ''y r*''. The extension to '''H''' will be complete when it is shown that ''f(q)'' = ''f(x'' − ''y r*''). Indeed, by hypothesis
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| :<math>u(x,y)=u(x,-y), \quad v(x,y) = -v(x,-y) \quad</math> so that one obtains
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| :<math>f(x-yr^*) = u(x,-y) + r^* v(x,-y) = u(x,y) + r \ v(x,y) = f(q).</math>
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| 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 differentiability. Therefore a direction-dependent derivative is necessary for functions of a quaternion variable.
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| == The Gâteaux derivative for quaternions ==
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| The [[Gâteaux derivative]] of a quaternionic function ''f(x)'' is given by
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| :<math>\partial f(x)(h)=\lim_{t\to 0 \in R}(t^{-1}(f(x+th)-f(x)))</math> | |
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| where ''h'' is a quaternion indicating the direction in which the derivative is to be taken. On the quaternions, the Gateaux derivative will always be [[linear mapping|linear]] in ''h'', so it may be expressed
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| as
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| :<math>\partial f(x)(h)= \sum_s \frac{{}_{(s)0}\partial f(x)}{\partial x} h
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| \frac{{}_{(s)1}\partial f(x)}{\partial x}.\,\!
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| </math>
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| The number of terms in the sum will depend on the function ''f''. The expressions
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| :<math>\frac{{}_{(s)p}\partial f(x)}{\partial x}, \ p=0,1\,\!</math>
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| are called components of the Gateaux derivative.
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| For the function ''f''(''x'') = ''axb'', the derivative is
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| :<math>\partial f(x)(h)=ahb\,\!</math> | |
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| and so the components are:
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| {| class="wikitable"
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| |-
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| | <math>\frac{{}_{(1)0}\partial axb}{\partial x}=a\,\!</math>
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| | <math>\frac{{}_{(1)1}\partial axb}{\partial x}=b\,\!</math>
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| |}
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| Similarly, for the function ''f''(''x'') = ''x''<sup>2</sup>, the derivative is
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| :<math>\partial f(x)(h)=xh+hx</math>
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| and the components are:
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| {| class="wikitable"
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| |-
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| | <math>\frac{{}_{(1)0}\partial x^2}{\partial x}=x\,\!</math>
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| | <math>\frac{{}_{(1)1}\partial x^2}{\partial x}=1\,\!</math>
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| |-
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| | <math>\frac{{}_{(2)0}\partial x^2}{\partial x}=1\,\!</math>
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| | <math>\frac{{}_{(2)1}\partial x^2}{\partial x}=x\,\!</math>
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| |}
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| Finally, for the function ''f''(''x'') = ''x''<sup>−1</sup>, the derivative is
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| :<math>\partial f(x)(h)=-x^{-1}hx^{-1}</math>
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| and the components are:
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| {| class="wikitable"
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| |-
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| | <math>\frac{{}_{(1)0}\partial x^{-1}}{\partial x}=-x^{-1}\,\!</math>
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| | <math>\frac{{}_{(1)1}\partial x^{-1}}{\partial x}=x^{-1}\,\!</math>
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| |}
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| ==References==
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| * [[Vladimir Arnold]] (1995) "The geometry of spherical curves and the algebra of quaternions", translated by [[Ian R. Porteous]], ''Russian Mathematical Surveys'' 50:1–68.
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| * C.A. Deavours (1973) "The Quaternion Calculus", [[American Mathematical Monthly]] 80:995–1008.
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| * [[Rudolf Fueter|R. Fueter]] (1936) "Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen" (in German), [[Commentarii Mathematici Helvetici]] 8: 371–378.
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| * Graziano Gentili, Catarina Stoppato & D.C. Struppa (2013) ''Regular Functions of a Quaternionic Variable'', Birkhäuser, ISBN 978-3-642-33870-0.
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| * P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", ''Proceedings of the [[Royal Irish Academy]]'', Section A 51: 67–85.
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| * K. Gürlebeck & W. Sprössig (1990) ''Quaternionic analysis and elliptic boundary value problems'', Birkhäuser ISBN 978-3-7643-2382-0 .
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| * W.R. Hamilton (1899) ''Elements of Quaternions'' v. I, edited by Charles Jasper Joly, "On differentials and developments of functions of quaternions", pages 430–64.
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| * R. Michael Porter (1998) [http://www.ams.org/journals/ecgd/1998-02-06/S1088-4173-98-00032-0/S1088-4173-98-00032-0.pdf Möbius invariant quaternion geometry], ''Conformal Geometry and Dynamics'' 2:89–196.
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| * A. Sudbery (1979) "Quaternionic Analysis", ''Mathematical Proceedings of the Cambridge Philosophical Society'' 85:199–225.
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| [[Category:Quaternions]]
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| [[Category:Functions and mappings]]
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| [[Category:Articles containing proofs]]
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Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. You can either install Word - Press yourself or use free services offered on the web today. This is a service where people write articles using a specific keyword you have given them. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster.
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Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Storing write-ups in advance would have to be neccessary with the auto blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. High Quality Services: These companies help you in creating high quality Word - Press websites. Some examples of its additional features include; code inserter (for use with adding Google Analytics, Adsense section targeting etc) Webmaster verification assistant, Link Mask Generator, Robots. Contact Infertility Clinic Providing One stop Fertility Solutions at:. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project.
Internet is not only the source for information, it is also one of the source for passive income. I'm a large fan of using Word - Press to create pretty much any sort of web page. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. If you loved this short article and you would like to receive much more information about wordpress backup plugin assure visit our web site. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010.