Poly-analytic function

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