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).

2 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).

3 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.

4 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. 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.

5 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.

6 In otherwords, there exists a Complex numbercsuch thatP(c) = 0. From this, it iseasy to deduce the following corollaries:1. 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.

7 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. 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.

8 (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!, which is defined for allz.

9 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.

10 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|< . 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.