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Polyanalytic Functions and Their Generalizations

Chapter
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 85)

Abstract

Polyanalytic functions emerged in the mathematical theory of elasticity: eighty years after the discovery of its basic equations, Kolossoff found that functions of the form φ(z) + Ψ(z), where φ and Ψ are analytic functions, can be an efficient tool for solving problems of the planar theory of elasticity. Functions of this form were later called bianalytic. Useful applications of this idea in mechanics are widely known from the remarkable investigations by Kolossoff, Muskhelishvili and their followers. The class of polyanalytic functions is an extension of the class of bianalytic functions. This survey is devoted to the former class.

Keywords

Entire Function Jordan Domain Essential Singularity Polyharmonic Function Condensation Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notations and Abbreviations

BA

bianalytic

CA

conjugate analytic

MA

meta-analytic

PA

polyanalytic (function)

OSCAR

an open simple curve (or arc) which is analytic and regular

H(G)

the class of functions holomorphic in a domain G

Hn(G), H(G)

the class of functions n-analytic (resp. conjugate analytic) in G

ℂ, \({\bar D_R}\)

the finite (resp. extended) complex plane

P

the p-dimensional complex space

Ck (G)

the class of functions u(x, y) + iv(x, y) with all partial derivatives in x, y up to the order k inclusive being continuous in the domain G

U(a), U0(a)

a neighborhood (resp. a punctured neighborhood) of a point a

D

the disk {z: |z| < 1}

T

the circumference {z: |z| = 1}

the closed disk {z: |z| ≤ R}

CR

the circumference {z: |z| = R}

Г(a; R)

the circumference {z: |z−a| = R}

D(a,R)

the disk {z: |z−a| < R}

deg wP

the exact degree of a polynomial P(z, w) with respect to w

deg w,zP

the same with respect to the pair of variables w and z

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