Complex VariablesComplex VariablesForForECON 397 ECON 397 MacroeconometricsMacroeconometricsSteve CunninghamSteve CunninghamOpen Disks or NeighborhoodsOpen Disks or NeighborhoodsDefinition. The set of all points z which Definition. The set of all points z which satisfy the inequality |satisfy the inequality |z z ––zz00|<|<ρρ, where , where ρρis is a positive real number is called an a positive real number is called an open diskopen diskor or neighborhood of zneighborhood of z0 0 . . Remark. The Remark. The unit diskunit disk, i.e., the neighborhood , i.e., the neighborhood ||zz|< 1, is of particular significance. |< 1, is of particular significance. 1Interior PointInterior PointDefinition. A point is called an Definition. A point is called an interior interior point of S point of S if and only if there exists at least one if and only if there exists at least one neighborhood of neighborhood of zz00which is completely which is completely contained in contained in SS. . Sz∈0z0SOpen Set. Closed Set.Open Set. Closed Set.Definition. If every point of a set Definition. If every point of a set SSis an interior is an interior point of point of SS, we say that , we say that SSis an is an open setopen set. . Definition. If Definition. If B(S) B(S) ⊂⊂S, S, i.e., if i.e., if SScontains all of its contains all of its boundary points, then it is called a boundary points, then it is called a closed setclosed set. . Sets may be neither open nor closed.Sets may be neither open nor closed.OpenClosedNeitherConnectedConnectedAn open set An open set SSis said to be is said to be connected connected if every pair if every pair of points of points zz11andandzz22in in S S can be joined by a can be joined by a polygonal line that lies entirely in polygonal line that lies entirely in SS. Roughly . Roughly speaking, this means that speaking, this means that SSconsists of a “single consists of a “single piece”, although it may contain holes. piece”, although it may contain holes. SSz1z2Domain, Region, Closure, Domain, Region, Closure, Bounded, CompactBounded, CompactAn open, connected set is called a An open, connected set is called a domain. domain. A A regionregionis a domain together with some, none, or all of its is a domain together with some, none, or all of its boundary points. The boundary points. The closure closure of a set of a set SSdenoted , is denoted , is the set of the set of SStogether with all of its boundary. Thustogether with all of its boundary. Thus..A set of points A set of points SSis is bounded bounded if there exists a positive if there exists a positive real number real number RRsuch that |such that |zz|<|<RRfor every for every z z ∈∈S.S.A region which is both closed and bounded is said A region which is both closed and bounded is said to be to be compact.compact.S)(SBSS ∪=Review: Real Functions of Real Review: Real Functions of Real VariablesVariablesDefinition. Let Definition. Let ΓΓ⊂⊂ℜℜ. A . A function ffunction fis a rule which is a rule which assigns to each element assigns to each element aa∈∈ΓΓone and only one one and only one element element b b ∈∈ΩΩ, , ΩΩ⊂⊂ℜℜ. We write . We write f: f: Γ→Γ→ΩΩ, or in , or in the specific case the specific case b = f(a)b = f(a), and call , and call b b ““the image of the image of aaunder under ff..””We call We call ΓΓ““the domain of definition of the domain of definition of f f ””or or simply simply ““the domain of the domain of f f ””. . We call We call ΩΩ““the range of the range of f.f.””We call the set of all the We call the set of all the imagesimagesof of ΓΓ, denoted , denoted f f ((ΓΓ), ), the image of the function the image of the function ff. We alternately call . We alternately call f f a a mapping mapping from from ΓΓto to ΩΩ..Real FunctionReal FunctionIn effect, a function of a real variable maps In effect, a function of a real variable maps from one real line to another.from one real line to another.ΓΩfComplex FunctionComplex FunctionDefinition. Complex function of a complex Definition. Complex function of a complex variable. Let variable. Let ΦΦ⊂⊂C. A C. A function f function f defined on defined on ΦΦis a rule which assigns to each is a rule which assigns to each zz∈∈ΦΦa a complex number complex number w. w. The number The number wwis called is called a a value of f at z value of f at z and is denoted by and is denoted by f(z)f(z), i.e., , i.e., w = f(z). w = f(z). The set The set ΦΦis called the is called the domain of definition of f. domain of definition of f. Although the domain of definition is often Although the domain of definition is often a domain, it need not be.a domain, it need not be.RemarkRemarkProperties of a realProperties of a real--valued function of a real valued function of a real variable are often exhibited by the graph of the variable are often exhibited by the graph of the function. But when function. But when w = f(z)w = f(z), where , where zzand and wware are complex, no such convenient graphical complex, no such convenient graphical representation is available because each of the representation is available because each of the numbers numbers zzand and wwis located in a plane rather than is located in a plane rather than a line. a line. We can display some information about the We can display some information about the function by indicating pairs of corresponding function by indicating pairs of corresponding points points z = z = ((x,yx,y) and ) and w = w = ((u,vu,v). To do this, it is ). To do this, it is usually easiest to draw the usually easiest to draw the z z and and w w planes planes separately.separately.Graph of Complex FunctionGraph of Complex Functionxuyvz-plane w-planedomain ofdefinitionrangew = f(z)Example 1Example 1Describe the range of the function f(z) = x2+ 2i, defined on (the domain is) the unit disk |z|≤ 1.Solution: We have u(x,y) = x2and v(x,y) = 2. Thus as z varies over the closed unit disk, u varies between 0 and 1, and v is constant (=2). Therefore w = f(z) = u(x,y) + iv(x,y) = x2+2i is a line segment from w = 2i to w = 1 + 2i.xyuvf(z)domainrangeExample 2Example 2Describe the function f(z) = z3for z in the semidisk given by |z|≤ 2, Im z ≥ 0.Solution: We know that the points in the sector of the semidisk from Arg z