Difference between revisions of "Poly-analytic function"
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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==== | ====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> | ||
Latest revision as of 19:59, 15 January 2024
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=14885
Poly-analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-analytic_function&oldid=14885
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article