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Complex Analysis: Supplementary Notes COMPLEX ANALYSIS: SUPPLEMENTARY NOTES PETE L. CLARK Contents Provenance 2 1. The complex numbers 2 1.1. Vista: The shadow of the real 2 1.2. Introducing C: something old and something new 4 1.3. Real part, imaginary part, norm, complex conjugation 7 1.4. Geometry of multiplication, polar form and nth roots 9 1.5. Topology of C 12 1.6. Sequences and convergence 16 1.7. Continuity and limits 19 1.8. The Extreme Value Theorem 21 2. Complex derivatives 21 2.1. Limits and continuity of functions f : A ⊂ C ! C 21 2.2. Complex derivatives 24 2.3. The Cauchy-Riemann Equations I 26 2.4. Harmonic functions 29 2.5. The Cauchy-Riemann Equations II 30 3. Some complex functions 31 3.1. Exponentials, trigonometric and hyperbolic functions 31 3.2. The extended complex plane 32 3.3. M¨obiustransformations 33 3.4. The Riemann sphere 37 4. Contour integration 37 4.1. Cauchy-Goursat 44 4.2. The Cauchy Integral Formula 46 4.3. The Local Maximum Modulus Principle 48 4.4. Cauchy's Integral Formula For Derivatives 50 4.5. Morera's Theorem 51 4.6. Cauchy's Estimate 52 4.7. Liouville's Theorem 52 4.8. The Fundamental Theorem of Algebra 53 4.9. Logarithms 53 4.10. Harmonic functions II 55 5. Some integrals 57 6. Series methods 60 6.1. Convergence, absolute convergence and uniform convergence 60 6.2. Complex power series 62 6.3. Analytic functions 65 6.4. The Identity Theorem 67 1 2 PETE L. CLARK 6.5. Laurent series 69 7. Isolated singularities 71 8. The residue calculus 76 8.1. The Residue Theorem 76 8.2. How to compute residues 78 8.3. Applications to integrals 80 8.4. Summation of series 83 9. The Riemann zeta function 86 References 87 Provenance These are lecture notes from my Fall 2017 Math 4150/6150 Complex Variables course. This is a one semester course at the advanced undergraduate level. The main prerequisite is some very elementary real analysis at the level of the Math 3100 (sequences and series) course that is taught at UGA. In places we also use a bit of linear algebra. I am not myself an expert in complex analysis, so I want to make clear that in writing these notes I am almost exclusively following other sources. The main text for the course was [BMPS], and overall we follow this rather closely, in particular in the ordering of topics. Perhaps it will be helpful to list the main differences between these notes and [BMPS]: • We introduce the complex logarithm rather late in the day, in x4.9. • We give a proof of the equivalence of connectedness and path-connectedness for open subsets of C, following a slick argument of [FB]. • Our discussion of contour integration is largely motivated by analogies to line integrals in mul- tivariable calculus. • We do not make use of the notion of homotopy of paths. (We certainly have nothing against it! But we think these ideas are better developed in either a second course on complex analysis or a first course in algebraic topology.) Our notion of simple connectedness is therefore in terms of simple closed curves being the boundaries of subdomains. We assume the Jordan-Schoenflies The- orem that every simple closed curve has a well-defined interior and exterior and that the interior is simply connected. Other than this, our discussion is self-contained and complete. • We include a proof of the Cauchy-Goursat Theorem, largely following [SS]. • We include a proof of Cauchy's Integral Theorem for Derivatives before our discussion of se- ries methods. On the other hand, we develop the basic theory of series a bit differently so that Cauchy's Integral Theorem for Derivatives also follows independently from this theory, whereas the development in [BMPS] uses Cauchy's Integral Theorem for the First Derivative. • We prove the Maximum Modulus Theorem more directly, without using harmonic functions. • Our discussion of computing Laurent series expansions is more detailed. • Our discussion on summation of series using residues is more detailed. Most of the additional details are taken from [MH, x4.4]. 1. The complex numbers 1.1. Vista: The shadow of the real. A complex number z is something of the form z = x + iy where x; y 2 R and i2 = −1. We notice that this is not quite a definition in a way that was deeply problematic for hundreds of years. Not to brag, but the very first thing we will actually do is COMPLEX ANALYSIS: SUPPLEMENTARY NOTES 3 totally resolve that problem. Just for a few minutes though, let's go with it. Complex numbers naturally arise while solving polynomial equations. The beginnings of this are taught in high school: the solution to the general quadratic equation ax2 + bx + c = 0 { for a; b; c real numbers with a 6= 0 { is p −b ± b2 − 4ac x = : 2a The quantity ∆ = b2 − 4ac is called the discriminant of the quadratic equation. Since the image of the squaring function R ! R; x 7! x2 is [0; 1), a real number has a real square root iff it is non-negative. So if ∆ > 0 the quadratic equation has two distinct roots. If ∆ = 0 the quadratic −b equation has just one root, r = 2a , but it occurs with multiplicity 2: that is, ax2 + bx + c = a(x − r)2: However, if ∆ < 0 then the formulap does not make sense in the real numbers. However, it does make sense in the complex numbers: ∆ = ipj∆j. Thus complex numbers are \born to give roots to all real quadratic equations." It turns out that the \equation solving ability" of the complex numbers is much more powerful than that. Soon enough we will see that for every nonzero complex number z and any positive integer n, there are exactly n complex numbers w such that wn = z. In particular, every nonzero complex number z has exactly two complex square roots, which are negatives of each other. From this and the same quadratic formula it follows that every quadratic equation az2 + bz + c = 0 with a; b; c 2 C and a 6= 0 has a root in z. Later on in the course we will see that much more is true: any nonconstant polynomial with complex coefficient has a complex root: this is the celebrated Fundamental Theorem of Algebra, first proved by Gauss. In many ways, the R world you have already met is just a shadow of the C world into which you're about to be thrust. Example 1.1. In the R-world, trigonometric functions cos x and sin x are very different from expo- nential functions ex: the former are periodic, hence bounded, whereas the latter grows exponentially fast. And yet there are some glimpses of a deeper relationship. For instance, consider one of the simplest differential equations: y00 = αy for α 2 R. What are the solutions? If α > 0, then they are exponential functions. If α < 0 they are trigonometric functions. Also consider: 1 X xn ex = ; n! n=0 1 1 X x2n X x2n+1 cos x = (−1)n ; sin x = (−1)n : (2n)! (2n + 1)! n=0 n=0 It looks like cos x and sin x are each \half of" ex { but something screwy is going on with the signs. But if we allow ourselves a complex variable, something amazing happens: first observe that 1 = i0 = i4 = i8 = :::; i = i1 = i5 = i9 = :::; −1 = i2 = i6 = i10 = :::; −i = i3 = i7 = i11 = :::: 4 PETE L. CLARK Then: 1 X (ix)n eix = = 1 + ix + i2x2=2! + i3x3=3! + i4x4=4! + ::: n! n=0 = (1 + i2x2=2! + i4x4=4! + i6x6=6! + :::) + i(x + i2x3=3! + i4x5=5! + i6x7=7! + :::) = (1 − x2=2! + x4=4! − x6=6! + :::) + i(x − x3=3! + x5=5! − x7=7! + :::) = cos x + i sin x: Thus, in the C-world the exponential function is periodic (its period is 2πi) and the trigonometric functions are unbounded. Later we will see that in the C-world every nonconstant \entire function" is unbounded! This relationship is not only true but profoundly useful. We give one example. Theorem 1.2. (Fresnel1 Integrals) We have Z 1 Z 1 rπ cos(x2)dx = sin(x2)dx = : 0 0 8 Complex analysis will reduce this integral to one familiar from multivariable calculus: 1 Z 2 p e−x dx = π: −∞ 1.2. Introducing C: something old and something new. In this section we introduce the complex field C. Recall that the real numbers R come endowed with binary operations of addition and multiplication + : R × R ! R; · : R × R ! R that satisfy the following properties: (P1) (Commutativity of +): For all x; y 2 R, we have x + y = y + x. (P2) (Associativity of +): For all x; y; z 2 R, we have (x + y) + z = x + (y + z). (P3) (Identity for +): There is 0 2 R such that for all x 2 R, we have x + 0 = x. (P4) (Inverses for +): For all x 2 R, there is y 2 R with x + y = 0. (P5) (Commutativity of ·): For all x; y 2 R, we have x · y = y · x. (P6) (Associativity of ·): For all x; y; z 2 R, we have (x · y) · z = x · (y · z). (P7) (Identity for ·): There is 1 2 R such that for all x 2 R, we have 1 · x = x. (P8) (Inverses for ·): For all x 2 R, if x 6= 0, then there is y 2 R with x · y = 1. (P9) (Distributivity of · over +): For all x; y; z 2 R, we have x · (y + z) = (x · y) + (x · z).
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