Ch.1:Complex Numbers Ch.1:Complex Numbers L1.6 Planar Sets L16 Pbrar Sets Planar Sets Open Disk(Neighborhood)and Open Set In the calculus of functions of a real variable,the main The set of all points that satisfy the inequality theorems are typically stated for functions defined on an z-z0<P interval,such as(0,1),(0,1,[0,1),0,1] where p is a positive number,is called an open disk or The interval can be interpreted as a segment in the x-axis in circular neighborhood of z0 z-plane A point zo which lies in a set S is called an interior point of A complex number is two-dimensional,hence for the functions S if there is some circular neighborhood of zo that is of a complex variable,the basic results are formulated for completely contained in S functions defined on sets that are 2-dimensional "domains"or If every point of a set S is an interior point of S.we say that "closed regions" S is an open set 白·0+之。·急，是2风C Ch.1:Complex Numbers Ch.1:Complex Numbers L16 Planar Scts 1.6 Ptanar Sets Domain Domain (Cont'd) An open set S is said to be connected if every pair of points The extension result to functions of two real variables: z1.z2 in S can be joined by a polygonal path that lies entirely Suppose u(z,y)is a real-valued function defined in a domain in S.Roughly speaking,this means that S consists of a D.If the first partial derivative of u satisfy "Single Piece" Ou du =0 An open connected set is called a domain 8x y For real variables,the derivative of the function equals zero at all points of D.then u =constant in D implies that this function is identically constant on the defined If D is merely assumed to be an open set(not connected) interval the theorem is no longer trueCh.1: Complex Numbers 1.6 Planar Sets Planar Sets In the calculus of functions of a real variable, the main theorems are typically stated for functions defined on an interval, such as (0, 1), (0, 1], [0, 1), [0, 1] The interval can be interpreted as a segment in the x-axis in z-plane A complex number is two-dimensional, hence for the functions of a complex variable, the basic results are formulated for functions defined on sets that are 2-dimensional ”domains” or ”closed regions” Ch.1: Complex Numbers 1.6 Planar Sets Open Disk (Neighborhood) and Open Set The set of all points that satisfy the inequality |z − z0| < ρ where ρ is a positive number, is called an open disk or circular neighborhood of z0 A point z0 which lies in a set S is called an interior point of S if there is some circular neighborhood of z0 that is completely contained in S If every point of a set S is an interior point of S, we say that S is an open set Ch.1: Complex Numbers 1.6 Planar Sets Domain An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S. Roughly speaking, this means that S consists of a ”Single Piece” An open connected set is called a domain For real variables, the derivative of the function equals zero implies that this function is identically constant on the defined interval Ch.1: Complex Numbers 1.6 Planar Sets Domain (Cont’d) The extension result to functions of two real variables: Suppose u(x, y) is a real-valued function defined in a domain D. If the first partial derivative of u satisfy ∂u ∂x = ∂u ∂y = 0 at all points of D, then u ≡constant in D If D is merely assumed to be an open set (not connected), the theorem is no longer true