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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  $  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.

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 [[#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====
<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" , ''Œuvres'' , '''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>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR>
</table>