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''of order  $  m $''

A complex function  $  w = u + iv $
of the real variables  $  x $
and  $  y $, 
or, which is equivalent, of the complex variables  $  z = x + iy $
and  $  \overline{z}\; = x - iy $, 
in a plane domain  $  D $
which can be represented as

$$ \tag{1 }
w  =  f( z, \overline{z}\; )  =  \sum_{k=0}^ { m-1} {\overline{z}\; } {}  ^ {k} f _ {k} ( z),
$$

where  $  f _ {k} ( z) $, 
$  k = 0 \dots m- 1 $, 
are complex-analytic functions in  $  D $. 
In other words, a poly-analytic function  $  w $
of order  $  m $
can be defined as a function which in  $  D $
has continuous partial derivatives with respect to  $  x $
and  $  y $, 
or with respect to  $  z $
and  $  \overline{z}\; $, 
up to order  $  m $
inclusive and which everywhere in  $  D $
satisfies the generalized Cauchy–Riemann condition:

$$ 

\frac{\partial   ^ {m} w }{\partial  {\overline{z}\; } {}  ^ {m} }
  =  0 .
$$

For  $  m = 1 $
one obtains analytic functions (cf. [[Analytic function]]).

For a function  $  u = u( x, y) $
to be the real (or imaginary) part of some poly-analytic function  $  w = u + iv $
in a domain  $  D $, 
it is necessary and sufficient that  $  u $
be a [[Poly-harmonic function|poly-harmonic function]] in  $  D $. 
One can transfer to poly-analytic functions certain classical properties of analytic functions, with appropriate changes (see [[#References|[1]]]).

A poly-analytic function of multi-order  $  m = ( m _ {1} \dots m _ {n} ) $
in the complex variables  $  z _ {1} \dots z _ {n} $
and  $  \overline{z}\; _ {1} \dots \overline{z}\; _ {n} $
in a domain  $  D $
of the complex space  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $, 
is a function of the form

$$ 
w  =  \sum _ {k _ {1} \dots k _ {n} = 0 } ^ { {m } _ {1} - 1 \dots m _ {n} - 1 } \overline{z}\; {} _ {1} ^ {k _ {1} } \dots \overline{z}\; {} _ {n} ^ {k _ {n} } f _ {k _ {1}  \dots k _ {n} } ( z _ {1} \dots z _ {n} ),
$$

where  $  f _ {k _ {1}  \dots k _ {n} } $
are analytic functions of the variables  $  z _ {1} \dots z _ {n} $
in  $  D $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.B. Balk,   M.F. Zuev,   "On polyanalytic functions"  ''Russian Math. Surveys'' , '''25''' :  5  (1970)  pp. 201–223  ''Uspekhi Mat. Nauk'' , '''25''' :  5  (1970)  pp. 203–226</TD></TR></table>

{{SourceattributionEOM|Poly-analytic function}}
[[Category:Mathematics]]