Advanced Complex Analysis - Harvard Mathematics Department

1 Advanced Complex AnalysisCourse Notes Harvard University Math 213aC. McMullenDecember 4, 2017 Contents1 Basic Complex Analysis ..22 The simply-connected Riemann surfaces .. 393 Entire and meromorphic functions .. 594 Conformal mapping .. 815 Elliptic functions and elliptic curves .. 108 ForwardComplex Analysis is a nexus for many mathematical fields, including:1. Algebra (theory of fields and equations);2. Algebraic geometry and Complex manifolds;3. Geometry (Platonic solids; flat tori; hyperbolic manifolds of dimen-sions two and three);4. Lie groups, discrete subgroups and homogeneous spaces ( (Z);5. Dynamics (iterated rational maps);6.)

2 Number theory and automorphic forms (elliptic functions, zeta func-tions);7. Theory of Riemann surfaces (Teichm uller theory, curves and their Ja-cobians);8. Several Complex variables and Complex manifolds;9. Real Analysis and PDE (harmonic functions, elliptic equations anddistributions).This course covers some basic material on both the geometric and analyticaspects of Complex Analysis in one :Background in real Analysis and basic differential topology(such as covering spaces and differential forms), and a first course in (These exercises are review.)1. LetT R3be the spherical triangle defined byx2+y2+z2= 1 andx,y,z 0. Let =z dxdz.(a) Find a smooth 1-form onR3such that =d.

3 (b) Define consistent orientations forTand T.(c) Using your choices in (ii), compute T and T directly, andcheck that they agree. (Why should they agree?)12. Letf(z) = (az+b)/(cz+d) be a M obius transformation. Show thenumber of rational mapsg: C Csuch thatg(g(g(g(g(z))))) =f(z)is 1, 5 or . Explain how to determine which alternative holds for Let anznbe the Taylor series for tanh(z) atz= 0.(a) What is the radius of convergence of this power series?(b) Show thata5= 2/15.(c) Give an explicit value ofNsuch that tanh(1) and N0anagreeto 1000 decimal places. Justify your Letf:U Vbe a proper local homeomorphism between a pair ofopen setsU,V C.

4 Prove thatfis a covering map. (Herepropermeans thatf 1(K) is compact wheneverK Vis compact.)5. Letf:C Cbe given by a polynomial of degree 2 or more. LetV1={f(z) :f (z) = 0} Cbe the set of critical values off, letV0=f 1(V1), and letUi=C Vifori= 0,1. Prove thatf:U0 U1is a covering Give an example whereU0/U1is a normal (or Galois) covering, ( 1(U0)) is a normal subgroup of 1(U1).1 Basic Complex analysisWe begin with an overview of basic facts about the Complex plane andanalytic Complex numbers will be denotedC. We let ,Hand Cdenote the unit disk|z|<1, the upper half plane Im(z)>0, and theRiemann sphereC { }. We writeS1(r) for the circle|z|=r, andS1forthe unit circle, each oriented counter-clockwise.

5 We also set = {0}andC =C {0}. Algebraic and analytic functionsThe Complex numbers are formally defined as the fieldC=R[i], wherei2= 1. They are represented in the Euclidean plane byz= (x,y) =x+ are two square-roots of 1 inC; the numberiis the one with positiveimaginary important role is played by the Galois involutionz7 z. We define|z|2=N(z) =zz=x2+y2. (Compare the case of a real quadratic field,whereN(a+b d) =a2 db2gives anindefiniteform.) Compatibility of|z|with the Euclidean metric justifies the identification ofCandR2. We alsosee thatzis a field: 1/z=z/|z|.It is also convenient to describe Complex numbers by polar coordinatesz= [r, ] =r(cos +isin ).

6 Herer=|z|and = argz R/2 Z. (The multivaluedness of argzrequirescare but is also the ultimate source of powerful results such as Cauchy sintegral formula.) We then have[r1, 1][r2, 2] = [r1r2, 1+ 2].In particular, the linear mapsf(z) =az+b,a6= 0, ofCto itself, preserveangles and formula should be provedgeometrically: in fact, it is a consequenceof the formula|ab|=|a||b|and properties of similar triangles. It can thenbe used to derive the addition formulas for sine and cosine (in Ahflors thereverse logic is applied).Algebraic critical feature of the Complex numbers is thatthey arealgebraically closed; every polynomial has a root.

7 (A proof will bereviewed below).Classically, the Complex numbers were introducing in the course of solv-ingrealcubic equations. Staring withx3+ax+b= 0 one can make aTschirnhaus transformation soa= 0. This is done by introducing a newvariabley=cx2+dsuch that yi= y2i= 0; even whenaandbare real,it may be necessary to chooseccomplex (the discriminant of the equationforcis 27b2+ 4a3.) It is negative when the cubic has only one real root;this can be checked by looking at the product of the values of the cubic atits max and an open set inCandf:U Ca sayfisanalyticiff (z) = limt 0f(z+t) f(z)t3exists for allz U. It is crucial here thattapproaches zero through arbitraryvalues inC.

8 Remarkably, this condition implies thatfis a smooth (C )function. For example, polynomials are analytic, as are rational functionsaway from their that anyreal linearfunction :C Chas the form (v) =av+ condition of analytic says thatDfz(v) =f (z)v; in other words, thevpart is make this point systematically, for a generalC1functionF:U Cwe definedFdz=12(dFdx+1idFdy)anddFdz=12(dFd x 1idFdy) We then haveDFz(v) =dFdzv+ can also write Complex -valued 1-formdFasdF= F+ F=dFdzdz+dFdzdzThusFis analytic iff F= 0; these are note that (d/dz)zn=nzn 1; a polynomialp(z,z) behaves as if thesevariables are of analytic and rational addition and multipli-cation we obtain naturally the polynomial functionsf(z) = n0anzn:C C.

9 The ring of polynomialsC[z] is an integral domain and aunique factorization domain, sinceCis a field. Indeed, sinceCisalgebraically closed, fact every polynomial factors into linear is useful to add the allowed value to obtain the Riemann sphere C=C { }. Then rational functions (ratiosf(z) =p(z)/q(z) of rel-atively prime polynomials, with the denominator not identically zero)determine rational mapsf:C C. The rational functionsC(z) arethe same as the field of fractions for the domainC[z]. We setf(z) = ifq(z) = 0; these points are called the rational and polynomial functions,the analytic functions includealgebraicfunctions such thatf(z) = z2+ 1.

10 A general algebraic functionf(z) satisfiesP(f) = N0an(z)f(z)n=0 for some rational functionsan(z); these arise, at least formally, when4one forms algebraic extension ofC(z). Such functions are generallymultivalued, so we must choose a particular branch to obtain an ana-lytic functions also arise when one solvesdifferential equations. Even equations with constant coefficients, likey +y= 0, can give rise to transcendental functions such as sin(z),cos(z) andez. Here are some useful facts about these familiar functionswhen extended toC:|exp(z)|= exp Rezcos(iz) = cosh(z)sin(iz) =isinh(z)cos(x+iy) = cos(x) cosh(y) isin(x) sinh(y)sin(x+iy) = sin(x) cosh(y) +icos(x) sinh(y).