Complex VariablesOpen Disks or NeighborhoodsInterior PointOpen Set. Closed Set.ConnectedDomain, Region, Closure, Bounded, CompactReview: Real Functions of Real VariablesReal FunctionComplex FunctionRemarkGraph of Complex FunctionExample 1Example 2PowerPoint PresentationSequenceLimit of a SequenceLimit of a FunctionLimits: InterpretationProperties of LimitsContinuityNote on ContinuityProperties of Continuous FunctionsDerivativesProperties of DerivativesAnalytic. Holomorphic.Rational Function.Slide 27Slide 28Testing for AnalyticityCauchy-Riemann Equations (1)Cauchy-Riemann Equations (2)Cauchy-Riemann Equations (3)Cauchy-Riemann Equations (4)Cauchy-Riemann Equations (5)Slide 35Slide 36Harmonic FunctionsHarmonic ConjugateExampleExample, ContinuedSlide 41Slide 42Complex ExponentialMore on ExponentialsComplex Complex VariablesVariablesForForECON 397 MacroeconometricsECON 397 MacroeconometricsSteve CunninghamSteve CunninghamOpen Disks or Open Disks or NeighborhoodsNeighborhoodsDefinition. The set of all points z which Definition. The set of all points z which satisfy the inequality |satisfy the inequality |z – zz – z00|<|<, where , where  is is a positive real number is called an a positive real number is called an open open diskdisk or or neighborhood of zneighborhood of z0 0 . . Remark. The Remark. The unit diskunit disk, i.e., the , i.e., the neighborhood |neighborhood |zz|< 1, is of particular |< 1, is of particular significance. significance. 1Interior PointInterior PointDefinition. A point is called an Definition. A point is called an interior point of S interior point of S if and only if there if and only if there exists at least one neighborhood of exists at least one neighborhood of zz00 which is completely contained in which is completely 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 SS is an interior is an interior point of point of SS, we say that , we say that SS is an is an open setopen set. . Definition. If Definition. If B(S) B(S)  S, S, i.e., if i.e., if SS contains 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 SS is said to be is said to be connected connected if if every pair of points every pair of points zz11 andand z z22 in in S S can be can be joined by a polygonal line that lies entirely joined by a polygonal line that lies entirely in in SS. Roughly speaking, this means that . Roughly speaking, this means that SS consists of a “single piece”, although it consists of a “single piece”, although it may contain holes. 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 regionregion is a domain together with some, none, or is a domain together with some, none, or all of its boundary points. The all of its boundary points. The closure closure of a set of a set SS denoted , is the set of denoted , is the set of SS together with all of its together with all of its boundary. Thusboundary. Thus . .A set of points A set of points SS is is bounded bounded if there exists a if there exists a positive real number positive real number RR such that | such that |zz|<|<RR for for every every z z  S. S.A region which is both closed and bounded is A region which is both closed and bounded is said to be said to be compact.compact.S)(SBSS Review: Real Functions of Real Review: Real Functions of Real VariablesVariablesDefinition. Let Definition. Let   . A . A function ffunction f is a rule is a rule which assigns to each element which assigns to each element aa   one one and only one element and only one element b b  , ,   . We . We write write f: f:  , or in the specific case , or in the specific case b = b = f(a)f(a), and call , and call b “b “the image of the image of aa under under ff.” .” We call We call  “the domain of definition of “the domain of definition of f ” f ” or or simply “the domain of simply “the domain of f ”. f ”. We call We call  “the “the range of range of f.f.” ” We call the set of all the We call the set of all the imagesimages of of , , denoted denoted f f ((), 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  to to ..Real FunctionReal FunctionIn effect, a function of a real variable In effect, a function of a real variable maps from one real line to another.maps from one real line to another.fComplex FunctionComplex FunctionDefinition. Complex function of a complex Definition. Complex function of a complex variable. Let variable. Let   C. A C. A function f function f defined on defined on  is a rule which assigns to each is a rule which assigns to each zz   a a complex number complex number w. w. The number The number ww is is called a called 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  is called the is called the domain of definition domain of definition of f. of f. Although the domain of definition is Although the domain of definition is often a domain, it need not be.often a domain, it need not be.RemarkRemarkProperties of a real-valued function of a real Properties of a real-valued function of a real variable are often exhibited by the graph of variable are often exhibited by the graph of the function. But when the function. But when w = f(z)w = f(z), where , where zz and and ww are complex, no such convenient graphical are complex, no such convenient graphical representation is available because each of representation is available because each of the numbers the numbers zz and and ww is located in a plane is located in a plane rather than a line. rather than a line. We can display some information about the We can display some information about the function by indicating pairs