The Great Picard Theorem Dennis Wahlström Bachelor’s Thesis, 15 Credits Bachelor of Mathematics, 180 Credits Spring 2018 Department of Mathematics and Mathematical Statistics 1 Abstract In this essay, we present a proof of the great Picard theorem by showing that a holomorphic function with an essential singularity attains infinitely many complex values in the vicinity of the singularity. Sammanfattning Vi kommer att presentera ett bevis på Picards stora sats genom att visa att en holomorf funktion med en väsentlig singularitet antar oändligt många komplexa värden i ett område av singulariteten. Contents Acknowledgements 3 1. Introduction 5 2. Complex Analysis 7 3. Conformal Mappings and Automorphism Groups 11 3.1. Automorphism Groups 14 3.2. Schwarz Reflection Principle 19 3.3. The Extended Plane and Carathéodory’s Theorem 20 4. A Modular Function 25 5. Covering Maps 33 6. The Great Picard Theorem 41 7. References 45 Acknowledgements Firstly, I would like to thank my supervisor, Per Åhag with my deepest gratitude for all his supports and advises during my difficult time, that was one the most challenging period of my life. Without Per, it would have been more difficult to be able to spell the word success. Secondly, many thanks to Aron Persson, Jesper Karlsson and Robin Törnkvist for took times to read and comment this essay. Lastly, I would like to thank my examiner, Olow Sande for his advises on this essay. 3 1. Introduction Complex analysis is one of the mathematical branches, and without any doubt it is an important mathematical field which has contributed to many study applications related to our daily life experience, such as in physics and engineering. The complex analysis is the analysis of functions of so called complex numbers, which include the square root of negative numbers. These numbers were first encountered in the first century of Anno Domini by Hero of Alexandria, at that time they were considered nonsensical due to the lack of awareness [7]. Even so, it took the works of many mathematicians before the theory of complex variables was commenced in the eighteenth century by Leonhard Euler [17]. The theory of complex variables was heired by the three pioneers Augustin-Louis Cauchy, Bernhard Riemann and Karl Weierstrass with distinct thought on the subject [17]. Cauchy’s view on the theory of complex variables was circular functions and the exponential functions, meanwhile Riemann focused on the geometry and Weierstrass had presented a vision on the convergent power series [17]. These visions have their own significative meaning in the today’s mathematics. We will focus ourself on one of the succeeders in the theory of complex variables, Charles Émile Picard. In the nineteenth century, Picard presented his thesis in an application of linear complexes where he mainly dealt with geometrical aspect and as the result, it became an important property to the Riccati’s equation [5]. Picard fully became a professor at the age of thirty and during his time as professor, he taught countless students e.g. in rational mechanics, differential and integral calculus, higher algebra and analysis [5]. Moreover, his personalities and teaching skills were acknowledged by many including by one of his PhD student Jacques Hadamard, who had chosen to described his advisor as: ”A notable feature about Picard’s scientific personality was the perfection of his teaching, one of the most, if not the most, marvelous I have experienced. One could say of it, thinking of what Mozart is reported to have said of one of his own works, that ’there was not one word too much, nor one word lacking’.” [5]. A few years after Picard obtained his professorship, he realised that the Casorati- Weierstrass theorem for a holomophic function with essential singularities could be strengthened, which today is known as the great Picard theorem [5, 17]. Theorem 6.5 (The Great Picard Theorem). If f has an essential singularity at z0, then with at most one exception, f attains every complex value infinitely many times in every neighborhood of z0. The original proof of the great Picard theorem can be found in [15,16]. It has many gen- eralisations in the area of analytic mappings to an extended plane for essential functions as well as for a punctured disk [11,14,18]. Besides these, it also has generalitions to e.g. Montel’s theorem, Schottky’s theorem and Bloch’s theorem [3, 19]. The prerequisites of this essay are a first course in complex analysis, a common prop- erty in calculus e.g. the real-differentiability and a course in linear algebra. Those who are familiar with algebraic topology and algebraic structure or advanced complex anal- ysis will have a greater advantage in pursuing the great Picard theorem in Section 6, 5 but we encourage to all the readers to explore the other sections filled with interesting proofs. The layout of this essay is as follows: first a review of complex analysis, and in Section 3 we shall define the automorphism group and construct the extended complex plane. Section 4 contains a construction of a modular function, and in Section 5 we will tie most of our results together preparing for Section 6, which is the great Picard theorem. The main theorems in this thesis are presented in Figure 1.1. This thesis is based on [20] with the focus on simplifying and clarifying for the readers who has not studied the advanced calculus course as it was one of prerequisites for the book. Moreover, a few definitions were alternatively taken from [9]. Theorem 6.5 (The Great Picard Theorem) Theorem 6.3 Theorem 6.4 Theorem 6.2 Theorem 6.1 Theorem 2.6, Theorem 4.10, Theorem 5.11, Corollary 5.12, Theorem 5.14, Theorem 5.17, Lemma 5.18, Theorem 5.19 Figure 1.1 – Architecture of the essay presented in a flowchart. 6 2. Complex Analysis The goal in this section is to reconstruct the prerequisites from complex analysis in the usage to construct conformal maps and modular functions. We will start off by introducing the concept of complex differentiable. Proposition 2.1. Suppose that V is an open subset of C, that f : V → C, and z ∈ V . Let u and v be the real and imaginary of f. Then f = u + iv is complex-differentiable at z, if, and only if, it is real-differentiable at z and the real and imaginary parts satisfy the Cauchy-Riemann equations ux(z) = vy(z), uy(z) = −vx(z) where x and y are the real part in R, and ux, uy vx and vy are the partial derivative. Throughout the essay we will denote V as an open set of the set of all complex numbers C. The Proposition 2.1 gives us authority to define the concept of holomorphic function, which is a complex function that is differentiable everywhere in the domain. Definition 2.2. Suppose that V ⊂ C is open and f : V → C. We say that f is holomorphic in V if f is complex differentiable at every point of V . The class of all functions holomorphic in V is denoted H(V ). There are various type of singularities, isolated singularity, removable singularity and essential singularity. By any means, the singularity is a point in which the mathematical object is not defined, e.g. differentiability fails. These singularities have their definition as following. Definition 2.3. Let f be a function in C and z ∈ C, suppose that f has an isolated singularity at z. The singularity is removable if f can be defined at z such that f becomes holomorphic in a neighborhood of z. The singularity is a pole if limw→z f(w) = ∞. Finally, an isolated singularity which is neither a pole nor a removable singularity is an essential singularity. Let’s introduce an open and a closed disk with z as its center where z ∈ C and a finite radius r > 0: D(z, r) = {w ∈ C : |z − w| < r} and D(z, r) = {w ∈ C : |z − w| ≤ r}. (2.1) Further through the essay, we will encounter a bounded holomorphic function on the boundary of its domain which can be divided into two different cases, a constant and a nonconstant holomorphic function. However, we will state these in form of two famous theorems that goes by the name of open mapping theorem with bounds and Maximum modulus theorem. 7 Lets start with the open mapping theorem with bounds. Theorem 2.4. Suppose that f ∈ H(V ) and its second-order complex derivative |f 00(z)| ≤ B for all z ∈ V , where B is a bound of V . Suppose that z0 ∈ V and first-order complex 0 0 2|f (z0)| derivative f (z0) 6= 0. If r > 0 is small enough that D(z0, r) ⊂ V and also r < B then D(f(z0), δ) ⊂ f(D(z0, r)) for Br2 δ = r|f 0(z )| − . 0 2 Proof. See e.g. in ([17], p. 256-257) or Theorem 5.7 in [20]. Before we proceed with the proof of the Maximum modulus theorem let’s define power series. Furthermore, we shall observe that the power series contains an important property to the analytical functions as well to the Maximum modulus theorem. Hence, the Maximum modulus theorem will be used frequently through essay, where it shows a great deal to handle the boundness of the holomorphic functions, which is what we needed in our proof of the great Picard theorem. P∞ j Definition 2.5. A series on the form j=0 aj(z − z0) is called a power series. The constants aj are the coefficients of the power series.