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| In [[mathematics]], the '''Cayley transform''', named after [[Arthur Cayley]], is any of a cluster of related things. As originally described by {{Harvtxt|Cayley|1846}}, the Cayley transform is a mapping between [[skew-symmetric matrix|skew-symmetric matrices]] and [[special orthogonal matrix|special orthogonal matrices]]. In [[complex analysis]], the Cayley transform is a [[conformal map]]ping {{Harv|Rudin|1987}} in which the image of the upper complex half-plane is the unit disk {{Harv|Remmert|1991|pp=82ff, 275}}. And in the theory of [[Hilbert space]]s, the Cayley transform is a mapping between [[linear operator]]s {{Harv|Nikol’skii|2001}}.
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| == Matrix map ==
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| Among ''n''×''n'' [[square matrix|square matrices]] over the [[real number|reals]], with ''I'' the identity matrix, let ''A'' be any [[skew-symmetric matrix]] (so that ''A''<sup>T</sup> = −''A''). Then ''I'' + ''A'' is [[invertible matrix|invertible]], and the Cayley transform
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| :<math> Q = (I - A)(I + A)^{-1} \,\!</math>
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| produces an [[orthogonal matrix]], ''Q'' (so that ''Q''<sup>T</sup>''Q'' = ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as <math> Q = (I + A)^{-1}(I - A)</math>. In fact, ''Q'' must have determinant +1, so is special orthogonal. Conversely, let ''Q'' be any orthogonal matrix which does not have −1 as an [[eigenvalue]]; then
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| :<math> A = (I - Q)(I + Q)^{-1} \,\!</math>
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| is a skew-symmetric matrix. The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing ''Q'' = ''A''<sup>c</sup> and ''A'' = ''Q''<sup>c</sup>.
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| This version of the Cayley transform is its own functional inverse, so that ''A'' = (''A''<sup>c</sup>)<sup>c</sup> and ''Q'' = (''Q''<sup>c</sup>)<sup>c</sup>. A slightly different form is also seen {{Harv|Golub|Van Loan|1996}}, requiring different mappings in each direction (and dropping the superscript notation):
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| :<math>\begin{align}
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| Q &{}= (I - A)^{-1}(I + A) \\
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| A &{}= (Q - I)(Q + I)^{-1}
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| \end{align}</math>
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| The mappings may also be written with the order of the factors reversed {{Harv|Courant|Hilbert|1989|loc=Ch.VII, §7.2}}; however, ''A'' always commutes with (μ''I'' ± ''A'')<sup>−1</sup>, so the reordering does not affect the definition.
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| === Examples ===
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| In the 2×2 case, we have
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| :<math>
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| \begin{bmatrix} 0 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 0 \end{bmatrix}
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| \lrarr
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| \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} .
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| </math>
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| The 180° rotation matrix, −''I'', is excluded, though it is the limit as tan <sup>θ</sup>⁄<sub>2</sub> goes to infinity.
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| In the 3×3 case, we have
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| :<math>
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| \begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix}
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| \lrarr
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| \frac{1}{K}
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| \begin{bmatrix}
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| w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\
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| 2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\
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| 2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2
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| \end{bmatrix} ,
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| </math>
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| where ''K'' = ''w''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>, and where ''w'' = 1. This we recognize as the rotation matrix corresponding to [[quaternion]]
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| :<math> w + \bold{i} x + \bold{j} y + \bold{k} z \,\!</math>
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| (by a formula Cayley had published the year before), except scaled so that ''w'' = 1 instead of the usual scaling so that ''w''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> = 1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan <sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup> = ''Q'').
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| === Other matrices ===
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| We can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, we may extend the definition further with choices of [[Hermitian adjoint|adjoint]] other than transpose or conjugate transpose.
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| Formally, the definition only requires some invertibility, so we can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example, we have
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| :<math>
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| \begin{bmatrix} 0 & -a & ab - c \\ 0 & 0 & -b \\ 0 & 0 & 0 \end{bmatrix}
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| \lrarr
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| \begin{bmatrix} 1 & 2a & 2c \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{bmatrix} .
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| </math> | |
| We remark that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.
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| == Conformal map ==
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| [[Image:Cayley transform in complex plane.png|thumb|right| 300px|Cayley transform of upper complex half-plane to unit disk]]
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| In [[complex analysis]], the Cayley transform is a [[mapping (mathematics)|mapping]] of the [[complex plane]] to itself, given by
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| :<math> \operatorname{W} \colon z \mapsto \frac{z-\bold{i}}{z+\bold{i}} . </math>
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| This is a [[linear fractional transformation]], and can be extended to an [[automorphism]] of the [[Riemann sphere]] (the [[complex plane]] augmented with a point at infinity).
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| Of particular note are the following facts:
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| * W maps the upper half plane of '''C''' [[conformal mapping|conformally]] onto the unit disc of '''C'''.
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| * W maps the real line '''R''' [[injective]]ly into the [[circle group|unit circle]] '''T''' (complex numbers of [[absolute value]] 1). The image of '''R''' is '''T''' with 1 removed.
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| * W maps the upper imaginary axis '''i''' <nowiki>[0, ∞)</nowiki> [[bijection|bijectively]] onto the half-open interval <nowiki>[−1, +1)</nowiki>.
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| * W maps 0 to −1.
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| * W maps the point at infinity to 1.
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| * W maps −'''i''' to the point at infinity (so W has a [[pole (complex analysis)|pole]] at −'''i''').
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| * W maps −1 to '''i'''.
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| * W maps both <sup>1</sup>⁄<sub>2</sub>(−1 + √3)(−1 + '''i''') and <sup>1</sup>⁄<sub>2</sub>(1 + √3)(1 − '''i''') to themselves.
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| == Operator map ==
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| An infinite-dimensional version of an [[inner product space]] is a [[Hilbert space]], and we can no longer speak of [[matrix (mathematics)|matrices]]. However, matrices are merely representations of [[linear operator]]s, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators.
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| :<math>\begin{align}
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| U &{}= (A - \bold{i}I) (A + \bold{i}I)^{-1} \\
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| A &{}= \bold{i}(I + U) (I - U)^{-1}
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| \end{align}</math>
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| Here the domain of ''U'', dom ''U'', is (''A''+'''i'''''I'') dom ''A''. See [[self-adjoint operator#Extensions of symmetric operators|self-adjoint operator]] for further details.
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| == See also ==
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| * [[Bilinear transform]]
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| * [[Extensions of symmetric operators]]
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| == References ==
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| * {{Citation
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| | last=Cayley
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| | first=Arthur
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| | author-link=Arthur Cayley
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| | year=1846
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| | title=Sur quelques propriétés des déterminants gauches
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| | journal=[[Journal für die reine und angewandte Mathematik]]
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| | volume=32
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| | pages=119–123
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| | url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D268141
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| | issn=0075-4102
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| }}; reprinted as article 52 (pp. 332–336) in {{Citation
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| | last=Cayley
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| | first=Arthur
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| | author-link=Arthur Cayley
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| | year=1889
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| | title=The collected mathematical papers of Arthur Cayley
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| | publisher=[[Cambridge University Press]]
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| | volume=I (1841–1853)
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| | pages=332–336
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| | isbn=<!-- none given -->
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| | url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000349
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| }}
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| * {{Citation
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| | last1=Courant
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| | first1=Richard
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| | author1-link=Richard Courant
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| | last2=Hilbert
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| | first2=David
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| | author2-link=David Hilbert
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| | title=Methods of Mathematical Physics
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| | volume=1
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| | edition=1st English
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| | publisher=Wiley-Interscience
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| | year=1989
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| | place=New York
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| | isbn=978-0-471-50447-4
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| }}
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| * {{Citation
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| | last1=Golub
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| | first1=Gene H.
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| | author1-link=Gene H. Golub
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| | last2=Van Loan
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| | first2=Charles F.
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| | author2-link=Charles F. Van Loan
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| | title=Matrix Computations
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| | edition=3rd
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| | publisher=Johns Hopkins University Press
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| | year=1996
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| | place=Baltimore
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| | isbn=978-0-8018-5414-9
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| }}
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| * {{Citation
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| | last=Nikol’skii
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| | first=N. K.
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| | contribution=Cayley transform
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| | contribution-url=http://www.encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=12556
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| | title=[[Encyclopaedia of Mathematics]]
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| | year=2001
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| | publisher=[[Springer-Verlag]]
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| | isbn=978-1-4020-0609-8<!-- uncertain, web page gives 1402006098, which does not validate -->
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| }}; translated from the Russian {{Citation
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| | editor-last=Vinogradov
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| | editor-first=I. M.
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| | editor-link=Ivan Matveyevich Vinogradov
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| | title=Matematicheskaya Entsiklopediya
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| | place=Moscow
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| | publisher=Sovetskaya Entsiklopediya
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| | year=1977
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| }}
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| * {{Citation<!-- courtesy of [[User:CSTAR|]] -->
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| | last=Remmert
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| | first=Reinhold
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| | author-link=Reinhold Remmert
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| | translator=Robert B. Burckel (trans.)<!-- template does not provide for this -->
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| | title=Theory of Complex Functions
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| | series=Graduate Texts in Mathematics
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| | volume='''122''' of ''Graduate Texts in Mathematics'' (''Readings in Mathematics'')<!-- compensate for lack of template "series" support -->
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| | year=1991
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| | publisher=[[Springer-Verlag]]
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| | place=New York
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| | isbn=978-0-387-97195-7
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| }}, translated by Robert B. Burckel from {{Citation
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| | unused_data=Grundwissen Mathematik 5
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| | last=Remmert
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| | first=Reinhold
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| | author-link=Reinhold Remmert
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| | title=Funktionentheorie I
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| | edition=2nd
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| | year=1989
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| | publisher=[[Springer-Verlag]]
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| | isbn=978-3-540-51238-7
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| }}
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| * {{Citation
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| | last=Rudin
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| | first=Walter
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| | author-link=Walter Rudin
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| | title=Real and Complex Analysis
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| | edition=3rd<!-- date was 1966, no edition given -->
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| | publisher=McGraw-Hill
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| | year=1987<!-- March 1 -->
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| | isbn=978-0-07-100276-9
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| }}
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| == External links ==
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| * {{PlanetMath
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| | urlname=CayleyTransform
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| | title=Cayley's parameterization of orthogonal matrices
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| | id=6535
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| }}
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| [[Category:Conformal mapping]]
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| [[Category:Transforms]]
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| [[ru:Преобразование Мёбиуса#Примеры]]
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