Complex Analysis Oral Exam study notes Notes transcribed by Mihai Nica Abstract. These are some study notes that I made while studying for my oral exams on the topic of Complex Analysis. I took these notes from parts of the textbook by Joseph Bak and Donald J. Newman [1] and also a real life course taught by Fengbo Hang in Fall 2012 at Courant. Please be extremely caution with these notes: they are rough notes and were originally only for me to help me study. They are not complete and likely have errors. I have made them available to help other students on their oral exams. Contents Introductory Stu 6 1.1. Fundamental Theorem of Algebra 6 1.2. Power Series 7 1.3. Fractional Linear Transformations 9 1.4. Rational Functions 10 1.5. The Cauchy-Riemann Equations 11 The Complex Numbers 13 Functions of the Complex Variable z 14 3.6. Analytic Polynomials 14 3.7. Power Series 14 3.8. Dierentiability and Uniqueness of Power Series 15 Analytic Functions 16 4.9. Analyticity and the Cauchy-Riemann Equations 16 4.10. The functions ez, sin z and cos z 16 Line Integrals and Entire Functions 18 5.11. Properties of the Line Integral 18 5.12. The Closed Curve Theorem for Entire Functions 19 Properties of Entire Functions 21 6.13. The Cauchy Integral Formula and Taylor Expansion for Entire Functions 21 6.14. Liouville Theorems and the Fundemental Theorem of Algebra 23 Properties of Analytic Functions 25 7.15. The Power Series Representation for Functions Analytic in a Disc 25 7.16. Analytic in an Arbitarty Open Set 26 7.17. The Uniqueness, Mean-Value and Maximum-Modulus Theorems 26 Further Properties 29 8.18. The Open Mapping Theorem; Schwarz' Lemma 29 8.19. The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reection Principle 30 Simply Connected Domains 32 9.20. The General Cauchy Closed Curve Theorem 32 9.21. The Analytic Function log z 32 Isolated Singularities of an Analytic Function 34 3 CONTENTS 4 10.22. Classication of Isolated Singularties; Riemann's Principle and the Casorati-Weirestrass Theorem 34 10.23. Laurent Expansions 35 Introduction to Conformal Mapping 38 11.24. Conformal Equivalence 38 11.25. Special Mappings 40 The Riemann Mapping Theorem 41 12.26. Conformal Mapping and Hydrodynamics 41 12.27. The Riemann Mapping Theorem 41 12.28. Connection to the Dirichlet problem for the Laplace Eqn 44 12.29. Complex Theorems to Memorize 45 Bibliography 46 Schwarz Lemma f << 1 in unit disk means f<<z and f'(0) << 1 in disc Open Mapping Theorem f(Open) is Open FLTs are Unique D->D map = 2 \pi i Mean Value Theorem f is equal to the average of in a circle Maximum Modulous Thm f has no interior maxima Minimum Modulous Thm Any interior minimum has |f|=0 Integral of (z-a)^-1 -> f n Cauchy Integral Formula Integral around a rectangle of a entire function vanishes CONTENTS 5 Integral Thm II Closed Curve Thm II M-L formula Basics of Line Integrals - -Uniform Convergence -Anti-derivative Behaviour analytic and f n Unif on Cpct Lims are Diff f uniformly on compact sets, then f analytic Rectangle Thm II Integral around a rectangle for the "secant" function vanishes Taylor Series Entire functions have convergent Taylor series Morera's Thm If integral vanishes along rectangles, then f is analytic Rectangle Thm Integral around a rectangle of a entire function vanishes Integral Thm Every analytic function has an anti-derivative Closed Curve Thm Integral around closed curve vanishes Basics of Power Series -Radius of Convergence -Differentiability -Uniqueness Analytic everywhere except a line + Continuous => Analytic Everywhere Schwarz Reflection Princip f real on axis can be extended by \bar{ f {\bar {z} } Introductory Stu From real life course by Fengbo Hang. 1.1. Fundamental Theorem of Algebra We can motivate the study of complex analysis by the fundamental theorem of algebra. This theorem says that, unlike real numbers, every n−th degree complex valued polynomial has n roots. Already this shows us that some things are much nicer in complex numbers than real numbers. Let us begin by dening the complex numbers as a 2-d vector space over R: Definition. Let i so that i2 = −1 and dene C = fa + ib : a; b 2 Cg. This is a vector space over with basis . We also equip ourselves with the natural p R f1; ig norm, ja + ibj = a2 + b2. Remark. To really be precise, one can check that the set of matrices a −b b a is a commutative 2-dimensional eld and make the identication 1 $ Id and 0 −1 i $ to precisely dene the complex numbers. Notice that this ma- 1 0 trix arises as the matrix for the operator by . When φz : C ! C φz(w) = zw a −b z = a + ib, the matrix for φ in the basis f1; ig is precisely . z b a Lemma. Let Pn k be any polynomial of degree . Then p(z) = k=0 akz n ≥ 1 p(z) has at least one root in C. Proof. Suppose by contradiction has no zeros. Let be such that p z0 2 C jp(z0)j > 0 is minimal. By translating, we may suppose without loss of generality that z0 = 0. We have then the expression for p: m m+1 p(z) = a0 + amz + O(z ) am m m+1 = a0 1 + z + O(z ) a0 Write am = Reiφ in polar notation, and then for a parameter t > 0, plugin z? = a0 i π−φ te m to get: ? am m m+1 jp(z )j = a0 1 + z + O(z ) a0 φ m π−φ m+1 = ja0j 1 + Re t e + O(t ) m m+1 = ja0j 1 − Rt + O(t ) ? But if we take t suciently small, we will get jp(z )j < ja0j = jp(0)j which contradicts the fact that z = 0 has minimal modulus. 6 1.2. POWER SERIES 7 Lemma. Let Pn k be any polynomial. Then has complex p(z) = k=0 akz p n roots. Proof. (By induction on n) The base case n = 1 is clear. The above lemma, along with the factoring algorithm to factor out roots reduces the degree by one. 1.2. Power Series Definition. A power series is a function f :Ω ! C where Ω ⊂ C is a function dened as an innite sum: 1 X n f(z) = an(z − z0) n=0 Remark. The disadvantage of power series is that, because of the way they are dened, there can be problems with convergence of the innite sum. For example the function 1 is dened when but it happens that 1 P n for . 1−z z 6= 1 1−z = z jzj < 1 Definition. We say that a function f :Ω ! C is C−dierentiable at a point z if the following limit exists: f(z + h) − f(z) f 0 (z) = lim h!0 h Here h is a complex number, and the limit is to be interpreted with the topology on C induced by the norm |·|. (This is the same topology as on R2) Remark. It turns out that C−dierentiable functions and power series are actually the same! We will see what this is later. Example. The usual tricks for dierentiation in R will work for C too. For example, all polynomials are dierentiable, and one can prove the quotient rule works too, so that rational functions are dierentiable too. The following special class of rational functions will be of interest to us, so we dene it below: Definition. A fractional linear transformation is a rational function of the form: az + b f(z) = cz + d a b Where 6= 0. c d We will now prove some results about power series. Lemma. Let P n be a power series and suppose that is so f(z) = cnz z1 2 C that the power series converges at (i.e. makes sense). Then is z1 f(z1) 2 C f(z) absolutely convergent for all z with jzj < jz1j. Proof. The proof is a simple comparison to a geometric series. Since f(z1) is convergent, we know that n as . In particular, this sequence must jcnz1 j ! 0 n ! 1 be bounded then, i.e. we have a constant so that n . But then n C jcnz1 j < C jcnz j < n C z and the convergence follows by comparison to a geometric series. z1 Corollary. For P n, there exists an so that is abso- f(z) = cnz R 2 R f(z) lutely convergent for all z with jzj < R and divergent for all z with jzj > R. 1.2. POWER SERIES 8 Proof. Just let R = sup fjzj : f(z) is convergentgand the result follows from the above lemma. Definition. The R above is often called the radius of convergence. Corollary. Power series are continuous inside their radius of convergence. P Proof. Recall the Weirstrass M-Test says if jfn(x)j < Mn and if Mn < 1, P then fn(x) is converging uniformly. Using this convergence test, along with the fact that a uniform limit of continuous functions is continuous we get the result. P n Theorem. [Power series are dierentiable] For a power series f(z) = cnz , we have that f is dierentiable with: 0 X n−1 f (z) = ncnz Proof. We introduce the notation: Pn k, Pn k, fn(z) = k=1 ckz gn(z) = k=1 kckz and P1 k−1. The problem becomes showing that is dierentiable g(z) = k=1 kckz f and f 0 = g.