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''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735002.png" />''
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A complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735003.png" /> of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735005.png" />, or, which is equivalent, of the complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735007.png" />, in a plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p0735008.png" /> which can be represented as
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{{TEX|auto}}
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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/p073/p073500/p0735009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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''of order  ''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350011.png" />, are complex-analytic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350012.png" />. In other words, a poly-analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350013.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350014.png" /> can be defined as a function which in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350015.png" /> has continuous partial derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350017.png" />, or with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350019.png" />, up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350020.png" /> inclusive and which everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350021.png" /> satisfies the generalized Cauchy–Riemann condition:
+
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
  
<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/p073/p073500/p07350022.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
= f( z, \overline{z}\; = \sum _ { k= } 0 ^ { m- }  1 {\overline{z}\; } {}  ^ {k} f _ {k} ( z),
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350023.png" /> one obtains analytic functions (cf. [[Analytic function|Analytic function]]).
+
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:
  
For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350024.png" /> to be the real (or imaginary) part of some poly-analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350025.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350026.png" />, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350027.png" /> be a [[Poly-harmonic function|poly-harmonic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350028.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350029.png" /> in the complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350031.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350032.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350034.png" />, is a function of the form
+
\frac{\partial  ^ {m} w }{\partial  {\overline{z}\; } {}  ^ {m} }
 +
  = 0 .
 +
$$
  
<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/p073/p073500/p07350035.png" /></td> </tr></table>
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For  $  m = 1 $
 +
one obtains analytic functions (cf. [[Analytic function|Analytic function]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350036.png" /> are analytic functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073500/p07350038.png" />.
+
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
 +
 
 +
$$
 +
= \sum _ {k _ {1} \dots k _ {n} = 0 } ^ { {m } _ {1} - 1 \dots m _ {n} - 1 } \overline{z}\; {} _ {1} ^ {k _ {1} } \dots \overline{z}\; {} _ {n} ^ {k _ {n} } f _ {k _ {1}  \dots k _ {n} } ( z _ {1} \dots z _ {n} ),
 +
$$
 +
 
 +
where    f _ {k _ {1}  \dots k _ {n} }
 +
are analytic functions of the variables    z _ {1} \dots z _ {n}
 +
in   D .
  
 
====References====
 
====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>
 
<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>

Revision as of 08:06, 6 June 2020


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 in D . One can transfer to poly-analytic functions certain classical properties of analytic functions, with appropriate changes (see [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

[1] 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
How to Cite This Entry:
Poly-analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-analytic_function&oldid=48231
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article