NOT FOR SALE OR DISTRIBUTION chapter 2

1 04455 CH02 Mathews 2010/11/20 8:07 page 53 #1 chapter 2complex functions OverviewThe last chapter developed a basic theory of complex numbers. For the next fewchapters, we turn our attention tofunctionsof complex numbers. They are de-fine din a similar way to functions of real numbers that you stu die din calculus;the only difference is that they operate on complex numbers rather than realnumbers. This chapter focuses primarily on very basic functions, their represen-tations, an dproperties associate dwith functions such as limits an will learn some interesting applications as well as some exciting new FUNCTIONS AND LINEAR MAPPINGSA complex-valued functionfof the complex variablezis a rule that assignsto each complex numberzin a setDone an donly one complex writew=f(z) an dcallwtheimage ofzunderf. A simple exampleof a complex-value dfunction is given by the formulaw=f(z)= setDis calle dthedomain off,an dthe set of all images{w=f(z):z D}is calle dtherange off.

2 When the context is obvious, we omit the phrasecomplex-valued, an dsimply refer to a functionf,or to a complex can define the domain to be any set that makes sense for a given rule,so forw=f(z)=z2,we coul dhave the entire complex plane for the domainD,or we might artificially restrict the domain to some set such asD=D1(0) ={z:|z|<1}.Determining the range for a function defined by a formula is notalways easy, but we will see plenty of examples later on. In some contextsfunctions are referre dto Section , we use dthe termdomainto indicate a connected open speaking about the domain of afunction, however, we mean only the setof points on which the function is defined. This distinction is worth noting, andcontext will make clear the use Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION . Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning.

3 LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION 04455 CH02 Mathews 2010/11/20 8:07 page 54 #2 54 chapter 2 Complex Functionsyxvuw = f(z) = u + ivu = u(x, y) v = v(x, y) mappingw=f(z).Just aszcan be expresse dby its real an dimaginary parts,z=x+iy, wewritef(z)=w=u+iv, whereuandvare the real an dimaginary parts ofw,respectively. Doing so gives us the representationw=f(z)=f(x, y)=f(x+iy)=u+ onxandy, they can be considered to be real-valuedfunctions of the real variablesxandy; that is,u=u(x, y) andv=v(x, y).Combining these ideas, we often write a complex functionfin the formf(z)=f(x+iy)=u(x, y)+iv(x, y).

4 (2-1)Figure illustrates the notion of a function (mapping) using these symbols. EXAMPLE (z)=z4in the formf(z)=u(x, y)+iv(x, y).SolutionUsing the binomial formula, we obtainf(z)=(x+iy)4=x4+4x3iy+6x2(iy)2+4x( iy)3+(iy)4= x4 6x2y2+y4 +i 4x3y 4xy3 ,so thatu(x, y)=x4 6x2y2+y4andv(x, y)=4x3y 4xy3. EXAMPLE the functionf(z)=zRe (z)+z2+ Im (z) in theformf(z)=u(x, y)+iv(x, y).SolutionUsing the elementary properties of complex numbers, it follows thatf(z)=(x iy)x+ x2 y2+i2xy +y= 2x2 y2+y +i(xy),so thatu(x, y)=2x2 y2+yandv(x, y)=xy. Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION . Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning.

5 LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION 04455 CH02 Mathews 2010/11/20 8:07 page 55 #3 Functions and Linear Mappings55 Examples an show how to fin du(x, y) andv(x, y) when a rulefor computingfis given. Conversely, ifu(x, y) andv(x, y) are two real-valuedfunctions of the real variablesxandy, they determine a complex-valued functionf(x, y)=u(x, y)+iv(x, y), an dwe can use the formulasx=z+z2andy=z z2ito fin da formula forfinvolving the variableszandz. EXAMPLE (z)=4x2+i4y2by a formula involving the reveals thatf(z)=4 z+z2 2+i4 z z2i 2=z2+2zz+z2 i z2 2zz+z2 =(1 i)z2+(2+2i)zz+(1 i) in the expression of a complex functionfmay be gives us the polar representationf(z)=f rei =u(r, )+iv(r, ),(2-2)whereuandvare real functions of the real variablesrand.

6 Remark a given functionf, the functionsuandvdefined here aredifferent from those defined by Equation(2-1)because Equation(2-1)involvesCartesian coordinates and Equation(2-2)involves polar coordinates. EXAMPLE (z)=z2in both Cartesian an dpolar the Cartesian form, a simple calculation givesf(z)=f(x+iy)=(x+iy)2= x2 y2 +i(2xy)=u(x, y)+iv(x, y)so thatu(x, y)=x2 y2, andv(x, y)= the polar form, we refer to Equation(1-39)to getf rei = rei 2=r2ei2 =r2cos 2 +ir2sin 2 =U(r, )+iV(r, ),so thatU(r, )=r2cos 2 , andV(r, )=r2sin 2 . Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION . Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning.

7 LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION 04455 CH02 Mathews 2010/11/20 8:07 page 56 #4 56 chapter 2 Complex FunctionsOnce we have defineduandvfor a functionfin Cartesian form, we must usedifferent symbols if we want to expressfin polar form. As is clear here, thefunctionsuandUare quite different, as course, if we are workingonly in one context, we can use any symbols we choose. EXAMPLE (z)=z5+4z2 6 in polar , using Equation(1-39)we obtainf(z)=f rei =r5(cos 5 +isin 5 )+4r2(cos 2 +isin 2 ) 6= r5cos 5 +4r2cos 2 6 +i r5sin 5 +4r2sin 2 =u(r, )+iv(r, ).We now look at the geometric interpretation of a complex function.

8 IfDisthe domain of real-valued functionsu(x, y) andv(x, y), the equationsu=u(x, y) andv=v(x, y)describe a transformation (or mapping) fromDin thexyplane into theuvplane,also calle dthewplane. Therefore, we can also consider the functionw=f(z)=u(x, y)+iv(x, y)to be a transformation (or mapping) from the setDin thezplane onto therangeRin thewplane. This idea was illustrated in Figure In the followingparagraphs we present some additional key ideas. They are staples for any kindof function, an dyou shoul dmemorize all the terms in bol a subset of the domainDoff, the setB={f(z):z A}is calledtheimageof the setA, andfis sai dto mapAontoB. The image of a singlepoint is a single point, an dthe image of the entire domain,D,is the range, mappingw=f(z) is sai dto be fromAintoSif the image ofAis containedinS. Mathematicians use the notationf:A Sto indicate that a illustrates a functionfwhose domain isDan dwhose range shaded areas depict that the function function alsomapsAintoR,and, of course, it imageof a pointwis the set of all pointszinDsuch thatw=f(z).

9 The inverse image of a point may be one point, several points, ornothing at all. If the last case occurs then the pointwis not in the range example, ifw=f(z)=iz,the inverse image of the point 1 is the singlepointi,becausef(i)=i(i)= 1,andiis theonlypoint that maps to case ofw=f(z)=z2,the inverse image of the point 1 is the set{i, i}. Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION . Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning.

10 LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION 04455 CH02 Mathews 2010/11/20 8:07 page 57 #5 Functions and Linear Mappings57xyuvDARBR angew = f(z) = u + ivDomain onto B;fmapsA into will learn in chapter 5 that ifw=f(z)=ez,the inverse image of the point0 is the empty set there is no complex numberzsuch thatez= inverse image of a set of points,S,is the collection of all points in thedomain that map , it is possible for the inverse image ofRto be a function as well, but the original function must have a special property:A functionfis sai dto beone-to-oneif it maps distinct pointsz1 =z2ontodistinct pointsf(z1) =f(z2).