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