COMPLEXANALYSIS - LTH

1 complex ANALYSISFall 2006 Christer BennewitzM nne ifr n h gre zoneranalytiska funktionersvaret nu dig finna l tap od dlighetens g ta?F. L fflerCopyrightc 2006 by Christer BennewitzPrefaceThese notes are basically a printed version of my lectures in COMPLEXANALYSIS at the University of Lund. As such they present a limited viewof any of the subject matters brought up, caused by the time constraintsone is faced by in a series of lectures. The core of the subject, presentedin Chapter 3, is very strongly influenced by the treatment in Ahlfors complex Analysis, one of the genuine masterpieces of the subject. Anyreader who wants to find out more is advised to read this prerequisites are in principle the mathematics coursesgiven in the first two semesters in Lund. Most importantly, this in-cludes a reasonably complete discussion of analysis in one and severalvariables and basic facts about series of functions including absoluteand uniform convergence.

2 A course in topology is also useful, but notessential. Primarily, a familiarity with the concept of a connected setis of , August 2006 Christer BennewitziContentsPrefaceiChapter 1. complex The complex number Polar form of complex Square Stereographic M bius Polynomials, rational functions and power series16 Chapter 2. Analytic Conformal mappings and Analyticity of power series; elementary Conformal mappings by elementary functions33 Chapter 3. complex Goursat s Local properties of analytic A general form of Cauchy s integral Analyticity on the Riemann sphere51 Chapter 4. Singular Laurent expansions and the residue Residue The argument principle65 Chapter 5. Harmonic Fundamental Dirichlet s problem79 Chapter 6. Entire Sequences of analytic Infinite Canonical Partial Hadamard s theorem96 Chapter 7. The Riemann mapping theorem101iiiivCONTENTSC hapter 8.

3 The Gamma function105 CHAPTER 1 complex The complex number systemRecall that agroup(G, )is a setGprovided with abinary opera-tion1 satisfying the following properties:(1)For all elementsx,yandz Gholds(x y) z=x (y z).(associative law)(2)There exists aneutral elemente Gwith the propertiesx e=e x=xfor everyx G.(3)Every elementx Ghas aninversex 1with the propertiesx x 1=x 1 x= that a set provided with an associative binaryoperation can haveat mostone neutral :Show that if the set has a left neutral element and a rightneutral element, they must that if a set has an associative binary opera-tion with neutral element, then any element of the set hasat :Show that if an element has a left inverse and a right inverse ,then these must group may also have the property(4)For all elementsxandy Gholdsx y=y x. (commutativelaw)in which case the group is called commutative orAbelian(after NielsHenrik Abel (1802 1829)).

4 Familiar examples of Abelian groups are(Z,+), the integers under ordinary addition;(R,+), the real num-bers under addition;(Rn,+), the set ofn-tuples of real numbers under(vector) addition; and(R\{0}, ), the non-zero real numbers undermultiplication. As an example of a non-Abelian group, consider theset of all rotations around lines through the origin in 3-dimensionalspace; the binary operation is the ordinary composition of maps. Thereader should check these examples carefully; in particular, find theneutral elements and inverses in these is, a map :G G G, so that for every pair of elementsx,yofG,there is a unique element ofGdenoted byx complex FUNCTIONSA field(F,+, )is a setFprovided with two binary operations+and , such that(F,+)is an Abelian group and, if0denotes the neutralelement of this group, also(F\{0}, )is an Abelian group. In additionthedistributive laws{(x+y) z=x z+y z,x (y+z) =x y+x for all elementsx,yandz F.}

5 It is usual to denote the neutralelement of(F\{0}, ) that in any fieldFholds0 x=x 0 = 0for allx F(as always,0denotes the neutral element of the group(F,+)). that a field does not have any non-zerodivi-sors of zero, , ifxy= 0, then eitherx= 0ory= examples of fields are(Q,+, ), the rational numbers underordinary addition and multiplication, and(R,+, ). We shall show, inthis section, that there is precisely one reasonable way of making theEuclidean plane into a field. By introducing Cartesian coordinatesthis plane may be identified with the Abelian group(R2,+), and wewill make this into a field by extending the usual multiplication of anelement ofR2by a real number. The resulting field is the fieldCofcomplex see how to make the definition, assume we have already managedto construct our fieldC. Then there is a multiplicative neutral element,which we will for the moment denote by1, to distinguish it from thereal number1.

6 We may identifyRwith the set of real multiples of1(explain!) and may therefore considerRas a subset ofC. Letebean element ofR2which is linearly independent of1, so that1,eis abasis inR2. Any elementz Cmay then be writtenz=x1+yewith real numbersxandy. In particular, there are real numbersaandbsuch thate2=a1+beso thatz2= (x2+ay2)1+ (2xy+by2)e(note that1 1=1,e 1=e). Now clearlyz2is real ify= 0(sinceactuallyzitself is, by the identification above). Butz2will also be realifx= b2y. We then getz2= (a+b24)y2. We can not havea+b24 0byExercise since then(z y a+b24)(z+y a+b24) = 0, but neitherof the factors is0unless theire-componenty= 0. Hencea+b24< we sety= 1/ (a+b24)we therefore getz2= , we have seen that if we can define a multiplication inR2which makes it into a field with addition being the ordinary vectoraddition, then there exists an element the square of which is 1(rather,the additive inverse of the multiplicative neutral element).

7 We pick onesuch element (we will see later that there are precisely two), denote THE complex NUMBER SYSTEM3byiand call it theimaginary unit. If we use1, ias a basis we maytherefore write any element in the plane asx1+yiwith realx,y. Forconvenience we will actually write itx+iyfrom now is important to note that we have not yet shown that it is possibleto make a field of the plane; we have just seen thatifit is possible,then we may identify thex-axis with the real numbers and they-axiswith the multiples of an element, the square of which is that if we calculate with symbolsx+iy, wherexandyare real numbers, according to the usual rules for adding andmultiplying numbers and in addition usei2= 1, then all the require-ments for a field are now on the field we have constructed is denoted byCandcalled the field of complex numbers. Note that the field of real numbersis anordered field. This means that we have a relation<defined amongthe real numbers such that(1)Ifxandy R, then exactly one ofx < y,y < xandx=yistrue.

8 (2)Sums and products of positive ( ,>0) numbers are have not introduced anything similar for the complex numbers forthe simple reason that itcan not be that in an ordered field squares of non-zeroelements are always>0. Use this to show that if it were possible tomakeCinto an ordered field, then both1>0and 1>0, and hencealso0>0, a a final note to this first section, the fact that the Euclidean planecan be made into a field is extremely useful in all areas of mathematicsand its applications. Since we live in a 3-dimensional (at least) world, itwould, from the point of view of applications, be very useful if we couldmake 3-dimensional space into a field as well. In the early part of thenineteenth century, this is exactly what the famous Irish mathematicianW. R. Hamilton tried, unsuccessfully, to to show that Hamilton was doomed to fail. Tosimplify things, you may require that the complex plane should bea 2-dimensional restriction of the 3-dimensional field.

9 Show that theexistence of divisors of zero can not be succeeded (1843) to introduce a multiplication inR4whichmakes this into a field, with the minor defect that the multiplicativegroup is not Abelian (such a structure is called askew field). Hamiltoncalled his structure thequaternions; this structure actually stronglyhints that it would be profitable, in physics, to consider the world 4-dimensional, with time as the fourth complex the set of symbolsx+iy+ju+kv, wherex,y,uandvare real numbers, and the symbolsi,j,ksatisfyi2=j2=k2= 1,ij= ji=k,jk= kj=iandki= ik=j. Showthat using these relations and calculating with the same formal rulesas in dealing with real numbers, we obtain a skew field; this is the setof Polar form of complex numbersIn the complex numberz=x+iythe real numberxis called thereal partofz,x= Rez, and the numberyis called theimaginarypartofz,y= Imz. There is of course nothing imaginary whateverabout the imaginary part; the reasons for this curious appellation arehistoric.

10 If we introduce the notationzfor the complex numberx iy,called thecomplex conjugateofz, we see thatRez=12(z+z)andImz=12i(z z). In particular,zis real ( , has imaginary part0)precisely ifz=z. Ifzhas real part0, so thatz= z, one callszpurely imaginary. We define theabsolute value|z|ofz=x+iyto be|z|= x2+y2. This is of course the ordinary length ofz, consideredas a vector in the plane, provided we draw1, ias orthonormal very useful observation is thatzz=|z| this and that for any complex numberszandwwe have(1)z+w=z+w,(2)zw=z w,(3)|zw|=|z||w|.It is worth remarking how one carries out division by a complexnumber. Since the complex numbers constitute a field, every non-zerocomplex number has a multiplicative inverse, , we can divide by it;namely, ifz6= 0andware complex numbers, then there is a uniquecomplex numberu, denotedwz, such thatzu=w. The question is,how does one write the quotient on the standard form as real part plusitimes imaginary part.