PART Ⅰ　Semi-analytic Function

Chapter Ⅰ　Definitions and Existence Theorems

1.1　Definitions

Let f（z）=u（x，y）+iv（x，y）be continuous in domain D.It is assumed that following f（z） is continuous.

Definition 1.1　A f（z） is called semi-analytic of the first kind at point（x，y），if ux and vy are continuous in a neighborhood of （x，y），and ux=vy at the point（x，y）.

Definition 1.2　A f（z） is called semi-analytic of the first kind in domain D，if f（z） is semi-analytic of the first kind for every point（x，y） in D.

Definition 1.3　A f（z） is called semi-analytic of the second kind at point （x，y），if uy and vx are continuous in a neighborhood of （x，y），and uy=-vx at the point（x，y）.

Definition 1.4　A f（z） is called semi-analytic of the second kind in domain D.if f（z） is semi-analytic of the second kind for every point （x，y） in D.

Definition 1.5　Let f（z） be semi-analytic in D，we consider a greater domain G（than D）which contains D.If F（z） is semi-analytic in G and F（z）=f（z） in D，then F（z） is called a semi-analytic extension of f（z） in G.

Definition 1.6　{D，f（z）} is called a semi-analytic element，where D is a domain and f（z） is semi-analytic in D.{D1，F1（z）}={D2，f2（z）}if and only if D1=D2，f1（z）=f2（z）.