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A representation of univalent functions (cf. [[Univalent function|Univalent function]]) that effect a conformal mapping of planar domains onto domains of a canonical form (for example, a disc with concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715502.png" />, included in one another, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715504.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715505.png" /> one assumes known a conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715506.png" /> onto some canonical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715507.png" />. From a known mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p0715509.png" /> onto a domain of canonical form one constructs such a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155012.png" /> is small. Under a continuous change of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155013.png" /> there arise in this way differential equations. The best known of these are the [[Löwner equation|Löwner equation]] and the Löwner–Kufarev equation. In the discrete case — for lattice domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155014.png" /> and a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155015.png" /> — the transition from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155016.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155018.png" />, is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see [[#References|[1]]]) and its generalizations (see [[#References|[2]]]). An equally important source of parametric representations are the Hadamard variations (see [[#References|[3]]], [[#References|[4]]]) for the Green functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155020.png" />, of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see [[#References|[5]]]). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.
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Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155021.png" /> is a collection of complex integers (a lattice domain) and that the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155022.png" /> is an extremal of the Dirichlet–Douglas functional
+
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 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155023.png" /></td> </tr></table>
+
A representation of univalent functions (cf. [[Univalent function|Univalent function]]) that effect a conformal mapping of planar domains onto domains of a canonical form (for example, a disc with concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains  $  Q _ {t} $,
 +
$  0 \leq  t < T $,
 +
included in one another,  $  Q _ {t  ^  \prime  } \subset  Q _ {t} $,
 +
$  0 \leq  t  ^  \prime  < t < T $.
 +
For  $  Q _ {0} $
 +
one assumes known a conformal mapping  $  f _ {0} $
 +
onto some canonical domain  $  B _ {0} $.
 +
From a known mapping  $  f _ {t} $
 +
of  $  Q _ {t} $
 +
onto a domain of canonical form one constructs such a mapping  $  f _ {t+ \epsilon }  $
 +
for  $  Q _ {t+ \epsilon }  $,
 +
where  $  \epsilon > 0 $
 +
is small. Under a continuous change of the parameter  $  t $
 +
there arise in this way differential equations. The best known of these are the [[Löwner equation|Löwner equation]] and the Löwner–Kufarev equation. In the discrete case — for lattice domains  $  Q _ {t} $
 +
and a natural number  $  t $—
 +
the transition from  $  f _ {t} $
 +
to  $  f _ {t + \epsilon }  $,
 +
$  \epsilon = 1 $,
 +
is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see [[#References|[1]]]) and its generalizations (see [[#References|[2]]]). An equally important source of parametric representations are the Hadamard variations (see [[#References|[3]]], [[#References|[4]]]) for the Green functions  $  G _ {t} ( z, z  ^  \prime  ) $,
 +
$  z, z  ^  \prime  \in Q _ {t} $,
 +
of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see [[#References|[5]]]). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.
  
in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155024.png" /> of all real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155026.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155027.png" />,
+
Suppose that  $  Q $
 +
is a collection of complex integers (a lattice domain) and that the Green function  $  G _ {z} ( z, z  ^  \prime  ) $
 +
is an extremal of the Dirichlet–Douglas functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
I _ {t} ( g)  = 2g( z  ^  \prime  ) + \sum _ { k= } 0 ^ { l }  \sum _ {z \in Q _ {0}  } \rho _ {k} ( t)  | \nabla _ {k} g( z) |  ^ {2}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155029.png" /></td> </tr></table>
+
in the class $  R _ {0} $
 +
of all real-valued functions  $  g( z) $
 +
on  $  Q $.
 +
Here  $  Q _ {0} = \{ {z } : {z, z- 1, z- i, z- 1- i \in Q } \} $,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155030.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155031.png" /> is the Kronecker symbol, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155033.png" />, is a certain collection of pairs of numbers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155034.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155036.png" /> or 1. To find an extremum of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155037.png" /> is a problem of quadratic programming. A comparison of its solutions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155039.png" /> gives the basic formula of invariant imbedding (Hadamard variation):
+
$$ \tag{1 }
 +
\left . \begin{array}{c}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
{\nabla _ {0} g( z)  = g( z) - g( z- 1- i), }
 +
\\
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155041.png" /> and the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155042.png" /> denotes the difference operators (1) in the second argument of the Green function. Knowing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155043.png" /> one can obtain step-by-step (recurrently) from (2) all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155045.png" />. By constructing the Green function, one obtains from the lattice analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155046.png" /> according to the equation of Cauchy–Riemann type
+
{\nabla _ {1} g( z)  = g( z- 1)- g( z- i), }
 +
\end{array}
 +
\right \}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155047.png" /></td> </tr></table>
+
$$
 +
\rho _ {k} ( 0)  \equiv  1,\  \rho _ {k} ( t+ 1)  = \rho _ {k} ( t) + N \delta _ {\zeta _ {t}  } ,
 +
$$
  
a univalent lattice quasi-conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155049.png" /> into the unit disc. Closest to the origin of coordinates is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155050.png" />. In the limit, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155051.png" />, the mapping is lattice conformal and the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155052.png" /> is a disc with concentric slits. The result is a continuous analogue of (2) (see [[#References|[6]]]). When all the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155053.png" /> are simply connected and the canonical domain is the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155054.png" />, one succeeds by using a fractional-linear automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155055.png" /> to represent the Green function in the explicit form
+
$  N $
 +
is a natural number,  $  \delta _ {\zeta _ {t}  } $
 +
is the Kronecker symbol, and  $  \zeta _ {t} = ( k _ {t} , z _ {t} ) $,
 +
$  t = 0 \dots T- 1 $,
 +
is a certain collection of pairs of numbers;  $  \{ {z _ {t} } : {t = 1 \dots T } \} $
 +
is the boundary of $  Q _ {t} $,  
 +
and  $  k _ {t} = 0 $
 +
or 1. To find an extremum of the functional  $  I _ {t} ( g) $
 +
is a problem of quadratic programming. A comparison of its solutions for  $  t $
 +
and $  t+ 1 $
 +
gives the basic formula of invariant imbedding (Hadamard variation):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155056.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
G _ {t+} 1 ( z, z  ^  \prime  )  = G _ {t} ( z, z  ^  \prime  ) -  
 +
\frac{1}{c _ {t} }
 +
\nabla _ {k _ {t}  } G _ {t} ( z _ {t} , z) \nabla _ {k _ {t}  } G _ {t} ( z _ {t} , z  ^  \prime  ),
 +
$$
  
in terms of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155057.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155058.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155059.png" /> with the normalization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155061.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155062.png" />.
+
where  $  c _ {t} = N  ^ {-} 1 - \nabla _ {k _ {t}  } \nabla _ {k _ {t}  }  ^  \prime  G _ {t} ( z _ {t} , z _ {t} ) > 0 $
 +
and the symbol  $  \nabla _ {k}  ^  \prime  $
 +
denotes the difference operators (1) in the second argument of the Green function. Knowing  $  G _ {0} ( z, z  ^  \prime  ) $
 +
one can obtain step-by-step (recurrently) from (2) all the functions  $  G _ {t} ( z, z  ^  \prime  ) $,
 +
$  t = 1 \dots T $.  
 +
By constructing the Green function, one obtains from the lattice analytic function  $  f _ {T} ( z) = G _ {T} ( z, z  ^  \prime  ) + iH _ {T} ( z, z  ^  \prime  ) $
 +
according to the equation of Cauchy–Riemann type
  
In terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155063.png" /> the Hadamard variation reduces to an ordinary (Löwner) differential equation. In comparison with the Hadamard variation this equation is considerably simpler; however, information on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155064.png" /> is only implicit in it — in terms of the control parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155065.png" />, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155066.png" /> is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.
+
$$
 +
(- 1) ^ {k} \nabla _ {1-} k H  = \rho _ {k} \nabla _ {k} G,
 +
$$
  
More general one-parameter families of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155068.png" />, not necessarily imbedded in one another, have also been treated (see [[#References|[7]]]). The equations arising in such parametric representations are called Löwner–Kufarev equations. There is also a modification of the Löwner and Löwner–Kufarev equations to the case when the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071550/p07155069.png" /> have a different kind of symmetry or other geometric peculiarities (see [[#References|[1]]]).
+
a univalent lattice quasi-conformal mapping  $  w =  \mathop{\rm exp} [ 2 \pi f( z)] $
 +
of $  Q _ {t} $
 +
into the unit disc. Closest to the origin of coordinates is the image of  $  z  ^  \prime  $.  
 +
In the limit, as  $  n \rightarrow \infty $,
 +
the mapping is lattice conformal and the image of  $  Q _ {T} $
 +
is a disc with concentric slits. The result is a continuous analogue of (2) (see [[#References|[6]]]). When all the domains  $  G _ {t} $
 +
are simply connected and the canonical domain is the unit disc  $  B $,
 +
one succeeds by using a fractional-linear automorphism of  $  B $
 +
to represent the Green function in the explicit form
 +
 
 +
$$
 +
G _ {t} ( z, z  ^  \prime  )  =   \mathop{\rm ln}  | 1- f _ {t} ( z) \overline{ {f _ {t} ( z  ^  \prime  ) }}\; | -  \mathop{\rm ln}  | f _ {t} ( z) - f _ {t} ( z  ^  \prime  ) |
 +
$$
 +
 
 +
in terms of the function  $  f _ {t} ( z) $
 +
mapping  $  Q _ {t} $
 +
onto  $  B $
 +
with the normalization  $  f( 0) = 0 $,
 +
$  0 \in Q _ {t} $
 +
for all  $  t \in [ 0, T) $.
 +
 
 +
In terms of  $  w = f _ {t} ( z) $
 +
the Hadamard variation reduces to an ordinary (Löwner) differential equation. In comparison with the Hadamard variation this equation is considerably simpler; however, information on the boundary of  $  Q _ {t} $
 +
is only implicit in it — in terms of the control parameter  $  \alpha ( t) =  \mathop{\rm arg}  f _ {t} ( z _ {t} ) $,
 +
because  $  f _ {t} ( z _ {t} ) $
 +
is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.
 +
 
 +
More general one-parameter families of domains  $  Q _ {t} $,
 +
$  0 \leq  t < T $,  
 +
not necessarily imbedded in one another, have also been treated (see [[#References|[7]]]). The equations arising in such parametric representations are called Löwner–Kufarev equations. There is also a modification of the Löwner and Löwner–Kufarev equations to the case when the domains $  Q _ {t} $
 +
have a different kind of symmetry or other geometric peculiarities (see [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Aleksandrov,  A.S. Sorokin,  "The problem of Schwarz for multiply connected circular domains"  ''Sib. Math. J.'' , '''13''' :  5  (1972)  pp. 671–692  ''Sibirsk. Mat. Zh.'' , '''13''' :  5  (1972)  pp. 971–1000</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Hadamard,  "Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées" , ''Oeuvres'' , '''2''' , CNRS  (1968)  pp. 515–642</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Hadamard,  "Leçons sur le calcul des variations" , '''1''' , Hermann  (1910)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Bellma,  E. Angel,  "Dynamic programming and partial differential equations" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Popov,  "Quantization of control systems"  ''Soviet Math. Dokl.'' , '''13''' :  6  (1972)  pp. 1668–1672  ''Dokl. Akad. Nauk. SSSR'' , '''207''' :  5  (1972)  pp. 1048–1050</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.P. Kufarev,  "On one-parameter families of analytic functions"  ''Mat. Sb.'' , '''13''' :  1  (1943)  pp. 87–118  (In Russian)  (English abstract)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Aleksandrov,  A.S. Sorokin,  "The problem of Schwarz for multiply connected circular domains"  ''Sib. Math. J.'' , '''13''' :  5  (1972)  pp. 671–692  ''Sibirsk. Mat. Zh.'' , '''13''' :  5  (1972)  pp. 971–1000</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Hadamard,  "Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées" , ''Oeuvres'' , '''2''' , CNRS  (1968)  pp. 515–642</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Hadamard,  "Leçons sur le calcul des variations" , '''1''' , Hermann  (1910)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Bellma,  E. Angel,  "Dynamic programming and partial differential equations" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Popov,  "Quantization of control systems"  ''Soviet Math. Dokl.'' , '''13''' :  6  (1972)  pp. 1668–1672  ''Dokl. Akad. Nauk. SSSR'' , '''207''' :  5  (1972)  pp. 1048–1050</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.P. Kufarev,  "On one-parameter families of analytic functions"  ''Mat. Sb.'' , '''13''' :  1  (1943)  pp. 87–118  (In Russian)  (English abstract)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>

Revision as of 08:05, 6 June 2020


A representation of univalent functions (cf. Univalent function) that effect a conformal mapping of planar domains onto domains of a canonical form (for example, a disc with concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains $ Q _ {t} $, $ 0 \leq t < T $, included in one another, $ Q _ {t ^ \prime } \subset Q _ {t} $, $ 0 \leq t ^ \prime < t < T $. For $ Q _ {0} $ one assumes known a conformal mapping $ f _ {0} $ onto some canonical domain $ B _ {0} $. From a known mapping $ f _ {t} $ of $ Q _ {t} $ onto a domain of canonical form one constructs such a mapping $ f _ {t+ \epsilon } $ for $ Q _ {t+ \epsilon } $, where $ \epsilon > 0 $ is small. Under a continuous change of the parameter $ t $ there arise in this way differential equations. The best known of these are the Löwner equation and the Löwner–Kufarev equation. In the discrete case — for lattice domains $ Q _ {t} $ and a natural number $ t $— the transition from $ f _ {t} $ to $ f _ {t + \epsilon } $, $ \epsilon = 1 $, is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see [1]) and its generalizations (see [2]). An equally important source of parametric representations are the Hadamard variations (see [3], [4]) for the Green functions $ G _ {t} ( z, z ^ \prime ) $, $ z, z ^ \prime \in Q _ {t} $, of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see [5]). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.

Suppose that $ Q $ is a collection of complex integers (a lattice domain) and that the Green function $ G _ {z} ( z, z ^ \prime ) $ is an extremal of the Dirichlet–Douglas functional

$$ I _ {t} ( g) = 2g( z ^ \prime ) + \sum _ { k= } 0 ^ { l } \sum _ {z \in Q _ {0} } \rho _ {k} ( t) | \nabla _ {k} g( z) | ^ {2} $$

in the class $ R _ {0} $ of all real-valued functions $ g( z) $ on $ Q $. Here $ Q _ {0} = \{ {z } : {z, z- 1, z- i, z- 1- i \in Q } \} $,

$$ \tag{1 } \left . \begin{array}{c} {\nabla _ {0} g( z) = g( z) - g( z- 1- i), } \\ {\nabla _ {1} g( z) = g( z- 1)- g( z- i), } \end{array} \right \} $$

$$ \rho _ {k} ( 0) \equiv 1,\ \rho _ {k} ( t+ 1) = \rho _ {k} ( t) + N \delta _ {\zeta _ {t} } , $$

$ N $ is a natural number, $ \delta _ {\zeta _ {t} } $ is the Kronecker symbol, and $ \zeta _ {t} = ( k _ {t} , z _ {t} ) $, $ t = 0 \dots T- 1 $, is a certain collection of pairs of numbers; $ \{ {z _ {t} } : {t = 1 \dots T } \} $ is the boundary of $ Q _ {t} $, and $ k _ {t} = 0 $ or 1. To find an extremum of the functional $ I _ {t} ( g) $ is a problem of quadratic programming. A comparison of its solutions for $ t $ and $ t+ 1 $ gives the basic formula of invariant imbedding (Hadamard variation):

$$ \tag{2 } G _ {t+} 1 ( z, z ^ \prime ) = G _ {t} ( z, z ^ \prime ) - \frac{1}{c _ {t} } \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z) \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z ^ \prime ), $$

where $ c _ {t} = N ^ {-} 1 - \nabla _ {k _ {t} } \nabla _ {k _ {t} } ^ \prime G _ {t} ( z _ {t} , z _ {t} ) > 0 $ and the symbol $ \nabla _ {k} ^ \prime $ denotes the difference operators (1) in the second argument of the Green function. Knowing $ G _ {0} ( z, z ^ \prime ) $ one can obtain step-by-step (recurrently) from (2) all the functions $ G _ {t} ( z, z ^ \prime ) $, $ t = 1 \dots T $. By constructing the Green function, one obtains from the lattice analytic function $ f _ {T} ( z) = G _ {T} ( z, z ^ \prime ) + iH _ {T} ( z, z ^ \prime ) $ according to the equation of Cauchy–Riemann type

$$ (- 1) ^ {k} \nabla _ {1-} k H = \rho _ {k} \nabla _ {k} G, $$

a univalent lattice quasi-conformal mapping $ w = \mathop{\rm exp} [ 2 \pi f( z)] $ of $ Q _ {t} $ into the unit disc. Closest to the origin of coordinates is the image of $ z ^ \prime $. In the limit, as $ n \rightarrow \infty $, the mapping is lattice conformal and the image of $ Q _ {T} $ is a disc with concentric slits. The result is a continuous analogue of (2) (see [6]). When all the domains $ G _ {t} $ are simply connected and the canonical domain is the unit disc $ B $, one succeeds by using a fractional-linear automorphism of $ B $ to represent the Green function in the explicit form

$$ G _ {t} ( z, z ^ \prime ) = \mathop{\rm ln} | 1- f _ {t} ( z) \overline{ {f _ {t} ( z ^ \prime ) }}\; | - \mathop{\rm ln} | f _ {t} ( z) - f _ {t} ( z ^ \prime ) | $$

in terms of the function $ f _ {t} ( z) $ mapping $ Q _ {t} $ onto $ B $ with the normalization $ f( 0) = 0 $, $ 0 \in Q _ {t} $ for all $ t \in [ 0, T) $.

In terms of $ w = f _ {t} ( z) $ the Hadamard variation reduces to an ordinary (Löwner) differential equation. In comparison with the Hadamard variation this equation is considerably simpler; however, information on the boundary of $ Q _ {t} $ is only implicit in it — in terms of the control parameter $ \alpha ( t) = \mathop{\rm arg} f _ {t} ( z _ {t} ) $, because $ f _ {t} ( z _ {t} ) $ is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.

More general one-parameter families of domains $ Q _ {t} $, $ 0 \leq t < T $, not necessarily imbedded in one another, have also been treated (see [7]). The equations arising in such parametric representations are called Löwner–Kufarev equations. There is also a modification of the Löwner and Löwner–Kufarev equations to the case when the domains $ Q _ {t} $ have a different kind of symmetry or other geometric peculiarities (see [1]).

References

[1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2] I.A. Aleksandrov, A.S. Sorokin, "The problem of Schwarz for multiply connected circular domains" Sib. Math. J. , 13 : 5 (1972) pp. 671–692 Sibirsk. Mat. Zh. , 13 : 5 (1972) pp. 971–1000
[3] J. Hadamard, "Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées" , Oeuvres , 2 , CNRS (1968) pp. 515–642
[4] J. Hadamard, "Leçons sur le calcul des variations" , 1 , Hermann (1910)
[5] R. Bellma, E. Angel, "Dynamic programming and partial differential equations" , Acad. Press (1972)
[6] V.I. Popov, "Quantization of control systems" Soviet Math. Dokl. , 13 : 6 (1972) pp. 1668–1672 Dokl. Akad. Nauk. SSSR , 207 : 5 (1972) pp. 1048–1050
[7] P.P. Kufarev, "On one-parameter families of analytic functions" Mat. Sb. , 13 : 1 (1943) pp. 87–118 (In Russian) (English abstract)

Comments

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
How to Cite This Entry:
Parametric representation of univalent functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametric_representation_of_univalent_functions&oldid=48127
This article was adapted from an original article by V.I. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article