2 Complex Functions and the Cauchy-Riemann …

1 2 Complex Functions and the Complex functionsIn one-variable calculus, we study functionsf(x) of a real variablex. Like-wise, in Complex analysis, we study functionsf(z) of a Complex variablez C(or in some region ofC). Here we expect thatf(z) will in generaltake values inCas well. However, it will turn out that some Functions arebetter than others. Basic examples of functionsf(z) that we have alreadyseen are:f(z) =c, wherecis a constant (allowed to be Complex ),f(z) =z,f(z) = z,f(z) = Rez,f(z) = Imz,f(z) =|z|,f(z) =ez. The func-tions f(z) = argz,f(z) = z, andf(z) = logzare also quite interesting,but they arenotwell-defined (single-valued, in the terminology of complexanalysis).What is a Complex valued function of a Complex variable? Ifz=x+iy,then a functionf(z) is simply a functionF(x,y) =u(x,y) +iv(x,y) of thetwo real variablesxandy. As such, it is a function (mapping) fromR2toR2. Here are some (z) =zcorresponds toF(x,y) =x+iy(u=x,v=y); (z) = z, withF(x,y) =x iy(u=x,v= y); (z) = Rez, withF(x,y) =x(u=x,v= 0, taking values just alongthe real axis); (z) =|z|, withF(x,y) = x2+y2(u= x2+y2,v= 0, takingvalues just along the real axis); (z) =z2, withF(x,y) = (x2 y2) +i(2xy) (u=x2 y2,v= 2xy); (z) =ez, withF(x,y) =excosy+i(exsiny) (u=excosy,v=exsiny).

2 Iff(z) =u+iv, then the functionu(x,y) is called thereal partoffandv(x,y) is called theimaginary partoff. Of course, it will not in general bepossible to plot the graph off(z), which will lie inC2, the set of orderedpairs of Complex numbers, but it is the set{(z,w) C2:w=f(z)}. Thegraph can also be viewed as the subset ofR4given by{(x,y,s,t) :s=u(x,y),t=v(x,y)}. In particular, it lies in a four-dimensional usual operations on Complex numbers extend to Complex Functions :given a Complex functionf(z) =u+iv, we can define Functions Ref(z) =u,1 Imf(z) =v,f(z) =u iv,|f(z)|= u2+v2. Likewise, ifg(z) is anothercomplex function, we can definef(z)g(z) andf(z)/g(z) for thosezfor whichg(z)6= of the most interesting examples come by using the algebraic op-erations ofC. For example, apolynomialis an expression of the formP(z) =anzn+an 1zn 1+ +a0,where theaiare Complex numbers, and it defines a function in the usualway. It is easy to see that the real and imaginary parts of a polynomialP(z)are polynomials inxandy.

3 For example,P(z) = (1 +i)z2 3iz= (x2 y2 2xy+ 3y) + (x2 y2+ 2xy 3x)i,and the real and imaginary parts ofP(z) are polynomials inxandy. Butgiven two (real) polynomial functionsu(x,y) andz(x,y), it is very rarely thecase that there exists a Complex polynomialP(z) such thatP(z) =u+ example, it is not hard to see thatxcannot be of the formP(z), nor can z. As we shall see later, no polynomial inxandytaking only real values foreveryz( 0) can be of the formP(z). Of course, sincex=12(z+ z)andy=12i(z z), every polynomialF(x,y) inxandyis also a polynomialinzand z, (x,y) =Q(z, z) = i,j 0cijzi zj,wherecijare Complex , while on the subject of polynomials, let us mention theFundamental Theorem of Algebra(first proved by Gauss in 1799): IfP(z) is a nonconstant polynomial, thenP(z) has a Complex root. In otherwords, there exists a Complex numbercsuch thatP(c) = 0. From this, it iseasy to deduce the following corollaries:1.

4 IfP(z) is a polynomial of degreen >0, thenP(z) can be factoredinto linear factors:P(z) =a(z c1) (z cn),for Complex numbersaandc1,.., Every nonconstant polynomialp(x) with real coefficients can be fac-tored into (real) polynomials of degree one or the first statement is a consequence of the fact thatcis a root ofP(z) ifand only if (z c) dividesP(z), plus induction. The second statement followsfrom the first and the fact that, for a polynomial withrealcoefficients, Complex roots occur in conjugate consequence of the Fundamental Theorem of Algebra is that, havingenlarged the real numbers so as to have a root of the polynomial equationx2+ 1 = 0, we are now miraculously able to find roots ofeverypolynomialequation, including the ones where the coefficients are allowed to be suggests that it is very hard to further enlarge the Complex numbers insuch a way as to have any reasonable algebraic properties. Finally, we shouldmention that, despite its name, the Fundamental Theorem of Algebra is notreally a theorem in algebra, and in fact some of the most natural proofs ofthis theorem are by using methods of Complex function can define a broader class of Complex Functions by dividing polynomi-als.

5 By definition, arational functionR(z) is a quotient of two polynomials:R(z) =P(z)/Q(z),whereP(z) andQ(z) are polynomials andQ(z) is not identically zero. Usingthe factorization (1) above, it is not hard to see that, ifR(z) is not actuallya polynomial, then it fails to be defined, roughly speaking, at the roots ofQ(z) which are not also roots ofP(z), and thus at finitely many points inC. (We have to be a little careful if there are multiple roots.)For Functions of a real variable, the next class of Functions we woulddefine might be the algebraic Functions , such as xor5 1 +x2 2 1 + , in the case of Complex Functions , it turns out to be fairly involvedto keep track of how to make sure these Functions are well-defined (single-valued) and we shall therefore not try to discuss them , there are Complex Functions which can be defined by power have already seen the most important example of such a function,ez= n=0zn/n!

6 , which is defined for allz. Other examples are, for instance,11 z= n=0zn,|z|< , to make sense of such expressions, we would have to discuss con-vergence of sequences and series for Complex numbers. We will not do sohere, but will give a brief discussion below of limits and continuity for com-plex Functions . (It turns out that, once things are set up correctly, thecomparison and ratio tests work for Complex power series.) Limits and continuityThe absolute value measures the distance between two Complex ,z1andz2are close when|z1 z2|is small. We can then define thelimitof a Complex functionf(z) as follows: we writelimz cf(z) =L,wherecandLare understood to be Complex numbers, if the distance fromf(z) toL,|f(z) L|, is small whenever|z c|is small. More precisely,if we want|f(z) L|to be less than some small specified positive realnumber , then there should exist a positive real number such that, if|z c|< , then|f(z) L|<.

7 Note that, as with real Functions , it doesnot matter iff(c) =Lor even thatf(z) be defined atc. It is easy to seethat, ifc= (c1,c2),L=a+biandf(z) =u+ivis written as a real andan part, then limz cf(z) =Lif and only if lim(x,y) (c1,c2)u(x,y) =aandlim(x,y) (c1,c2)v(x,y) =b. Thus the story for limits of Functions of a complexvariable is the same as the story for limits of real valued Functions of thevariablesx,y. However, a real variablexcan approach a real numberconlyfrom above or below (or from the left or right, depending on your point ofview), whereas there are many ways for a Complex variable to approach acomplex , limits of sequences, convergent series and power series can bedefined for Functions of a real variable, a functionf(z) iscontinuousatciflimz cf(z) =f(c).In other words: 1) the limit exists; 2)f(z) is defined atc; 3) its value atcis the limiting value. A functionf(z) iscontinuousif it is continuous at allpoints where it is defined.

8 It is easy to see that a functionf(z) =u+ivis continuous if and only if its real and imaginary parts are continuous, andthat the usual functionsz, z,Rez,Imz,|z|,ezare continuous. (We have tobe careful, though, about Functions such as argzor logzwhich are notwell-defined.) All polynomialsP(z) are continuous, as are all two-variablepolynomial Functions inxandy. A rational functionR(z) =P(z)/Q(z) withQ(z) not identically zero is continuous where it is defined, at the finitelymany points where the denominatorQ(z) is not zero. More generally, iff(z) andg(z) are continuous, then so (z), wherecis a constant; (z) +g(z); (z) g(z); (z)/g(z), where defined ( whereg(z)6= 0).5. (g f)(z) =g(f(z)), the composition ofg(z) andf(z), where Complex derivativesHaving discussed some of the basic properties of Functions , we ask nowwhat it means for a function to have acomplexderivative. Here we willsee something quite new: this is very different from asking that its real andimaginary parts have partial derivatives with respect toxandy.

9 We willnot worry about the meaning of the derivative in terms of slope, but onlyask that the usual difference quotient functionf(z) iscomplex differentiableatciflimz cf(z) f(c)z cexists. In this case, the limit is denoted byf (c). Making the change ofvariablez=c+h,f(z) is Complex differentiable atcif and only if the limitlimh 0f(c+h) f(c)hexists, in which case the limit is againf (c). A function iscomplex differen-tiableif it is Complex differentiable at every point where it is defined. Forsuch a functionf(z), the derivative defines a new function which we writeasf (z) orddzf(z).For example, a constant functionf(z) =Cis everywhere Complex differ-entiable and its derivativef (z) = 0. The functionf(z) =zis also complexdifferentiable, since in this casef(z) f(c)z c=z cz c= (z) = 1. But many simple Functions do not have Complex example, considerf(z) = Rez=x. We show that the limitlimh 0f(c+h) f(c)h5does not exist for anyc.

10 Letc=a+bi, so thatf(c) =a. First considerh=ta real number. Thenf(c+t) =a+tand sof(c+h) f(c)h=a+t at= if the limit exists, it must be 1. On the other hand, we could useh= this case,f(c+it) =f(c) =a, andf(c+h) f(c)h=a ait= approachingcalong horizontal and vertical directions has given twodifferent answers, and so the limit cannot exist. Other simple functionswhich can be shown not to have Complex derivatives are Imz, z, and|z|.On the bright side, the usual rules for derivatives can be checked to hold:1. Iff(z) is Complex differentiable, then so iscf(z), wherecis a constant,and (cf(z)) =cf (z);2. (Sum rule) Iff(z) andg(z) are Complex differentiable, then so isf(z)+g(z), and (f(z) +g(z)) =f (z) +g (z);3. (Product rule) Iff(z) andg(z) are Complex differentiable, then so isf(z) g(z) and (f(z) g(z)) =f (z)g(z) +f(z)g (z);4. (Quotient rule) Iff(z) andg(z) are Complex differentiable, then so isf(z)/g(z), where defined ( whereg(z)6= 0), and(f(z)g(z)) =f (z)g(z) f(z)g (z)g(z)2;5.