Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 63', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 402 Functions of Complex Variable of D with its boundary is called a closed domain and is denoted by D. The positive sense of the boundary is defined to be the sense for which the domain lies to the left of the boundary. The boundary of a domain can consist of finitely many closed curves segments and points the curves and cuts are assumed to be piecewise smooth. The simplest examples of domains are neighborhoods of points on the complex plane. A neighborhood of a point a on the complex plane is understood as the set of points z such that z - a R . the interior of the disk of radius R 0 centered at the point a. The extended complex plane is obtained by augmenting the complex plane with the fictitious point at infinity. A neighborhood of the point at infinity is understood as the set of points z such that z R including the point at infinity itself . If to each point z of a domain D there corresponds a point w resp. a set of points w then one says that there is a single-valued resp. multivalued function w f z defined on the domain D. If we set z x iy and w u iv then defining a function w f z of the complex variable is equivalent to defining two functions Re f u u x y and Im f v v x y of two real variables. If the function w f z is single-valued on D and the images of distinct points of D are distinct then the mapping determined by this function is said to be schlicht. The notions of boundedness limit and continuity for single-valued functions of the complex variable do not differ from the corresponding notions for functions of two real variables. . Differentiability and analyticity. Let a single-valued function w f z be defined in a neighborhood of a point z. If there exists a limit f Z h - f z . lim------------------ fz z h 0 h then the function w f z is said to be differentiable at the point z and fz z is called its derivative at the point z. Cauchy-Riemann conditions. If the functions u x y Re f z and v x y Im f z are differentiable at a point x
