Difference between revisions of "Poly-analytic function"
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− | + | ''of order '' | |
− | + | 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|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> |
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 |
Poly-analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-analytic_function&oldid=48231