= 0. Theorem (Maximum Principle) If a holomorphic function f defined on a domain D C takes the maximum max z D f(z), then f is constant.

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1 38 CHAPTER 3. HOLOMORPHIC FUNCTIONS Maximum Principle Maximum Principle was first proved for harmonic functions (i.e., the real part and the imaginary part of holomorphic functions) in Burkhardt s textbook in 897. It is not clear who was the first to prove this theorem for holomorphic functions. At least, in 92, C. Carathéodory gave his simple proof of the Schwarz lemma by means of the maximum principle for holomorphic functions. 26 By the point of view of modern mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. In fact, a harmonic function is a solution of the Laplace equation: 2 u + 2 u = 0. x 2 y 2 Theorem (Maximum Principle) If a holomorphic function f defined on a domain D C takes the maximum max z D f(z), then f is constant. Proof Suppose f(z 0 ) = max z D f(z) for some point z 0 D. Take a disk (z 0, r) D and write f(z) = a n (z z 0 ) n, z (z 0 ; r). By the theorem above, n=0 a n 2 r 2n = 2π f(re iθ ) 2 dθ so that a n= n=0 a n 2 r 2n 2π max z D f(z) 2 dθ = 2π f(z 0 ) 2 dθ = 2π a 0 2 dθ = a 0 2. It implies that a n = 0, n, i.e., f(z) = a 0 is constant. Liouville s Theorem The Cauchy inequality immediately imply the following Liouville s Theorem. Theorem 3.5. (Liouville s Theorem) Let f : C C be a bounded holomorphic function. Then f = constant. Proof: Write f(z) = n=0 a nz n, z C. By the condition, there exists a constant M > 0 such that f(z) M, z C. 26 R. Remmert, Theorey of Complex Functions, GTM 22, Springer, 99, p.259.

2 3.5. CAUCHY INTEGRAL FORMULA 39 For any n, we apply Theorem (i) to get a n 2πr n M 2πr n 2π 0 2π 0 f(re iθ ) dθ dθ = M r n 0, as r +. Then a n = 0, n, i.e., f(z) = a 0 = constant. [Example] Let f be a holomorphic function defined on C satisfying Re(f) M < +, z C. Let F(z) := e f(z) be a holomorphic function defined on C. Then F(z) = e f(z) = e Re(f(z)) < e M, z C. By applying Louville s Theorem, it follows that f = constant. Historic Remarks on Liouville s Theorem In 844, Louville presented a paper to the Academy on doubly periodic functions where he set up the following general principle : If such a function is doubly periodic, and if one recognizes that it never becomes infinite, one can, from this alone, affirm that it reduces to a constant. Cauchy quickly recognize the analogy between Liouville s general principle and his own results and at the very next meeting of the Academy, he rushed to present a paper where he recalled some of his old theorems on residues and promised to show in future papers how they were related to Liouville s principle which he stated in apparently more general terms: If a function f(z) of a real or imaginary variable z always remains continuous and consequently always finite it simply reduces to a constant. 27 Liouville ( ) put this theorem in his lecture in 847 and published it in 879. Since Liouville s theorem is a consequence of Cauchy s Integral Formula, the credit may belong to Cauchy. Remmert wrote: the (Liouville) theorem originated with Cauchy, who derived it in 844 in his note Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003, p R. Remmert, Theorey of Complex Functions, GTM 22, Springer, 99, p.246.

3 40 CHAPTER 3. HOLOMORPHIC FUNCTIONS Fundamental Theorem of Algebra Theorem (Fundamental Theorem of Algebra) If f(z) = a N z N + a N z N a z + a 0 constant, then there exists some z 0 C such that f(z 0 ) = 0. Proof Suppose that f(z) 0, z C. Then g(z) := is a holomorphic function defined f(z) on C. If we can show g(z) is bounded on C, (3.32) then by Liouvilles Theorem, g = constant so that f = constant, which is a contradiction. In fact, g(z) = = = f(z) a N z N + a N z N a z + a 0 z N an + a N a 0 z z N z N ( a N a N a 0 ) z z N 2 z N( a N a N ) = z N a N 2 R 0 a N, z R 0. 2 Here we find a constant R 0 > 0 such that when z R 0, a N a 0 z z N < a N 2. Then for any z C, So (3.32) is proved. { } 2 g(z) max R 0 a N, max z R 0 f(z) <. Another proof for Fundamental Theorem of Algebra: Let f(z) = z n +a n z n +...+a z+a 0 be a polynomial of degree n and assume that f(z) 0 for all z C.

4 3.5. CAUCHY INTEGRAL FORMULA 4 By Cauchy s integral theorem we have dz zf(z) = 2πi f(0) 0 z =r where the circle is traversed counterclockwise. On the other hand, dz zf(z) 2πr max z=r zf(z) = 2π min z=r f(z) 0 z=r as r 0 (since f(z) z n a n z... a 0 z n ), which is a contradiction.. 29 Historic Remark on the Fundamental Theorem of Algebra Peter Roth of Nurnberg, in his book Arithmetica Philosophica (published in 608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, a Flemish mathematician, in his book L invention nouvelle en l Algébre (published in 629), asserted that a polynomial equation of degree n has n solutions, but he did not state that the solutions are of the form a + ib with a, b real. Furthermore, he added that his assertion holds unless the equation is incomplete, by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation x 4 = 4x 3, although incomplete, has four solutions (counting multiplicities): (twice), + i 2, and i 2. In 637, Descartes also clearly stated this theorem. First version of the fundamental theorem of algebra is for polynomial with real coefficients. For such a polynomial f, it was believed that f can be written as a product of polynomials with real coefficients whose degree is either or 2. However, in 702 Leibniz claimed that such a theorem cannot be true by giving the following counterexample: x 4 +a 4 could never be written as a product of two real quadratic factors. (His mistake came in not realizing that i could be written in the form a+bi where a and b are real. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x 4 4x 3 + 2x 2 +4x+4, but he got a letter from Euler in 742 in which he was told that his polynomial happened to be equal to ( x 2 (2 + α)x α )( x 2 (2 + α)x α ) 29 A. R. Schep, A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra, Am Math Monthly 6, Jan 2009,

5 42 CHAPTER 3. HOLOMORPHIC FUNCTIONS where α is the square root of 4+2 7, and Euler also showed that Leibniz s counterexample was incorrect: x 4 + a 4 = (x 2 + a 2x + a 2 )(x 2 a 2 + a 2 ). A first attempt at proving the theorem was made by d Alembert in 746, but his proof was incomplete. Other attempts were made by Euler (749), de Foncenex (759), Lagrange (772), and Laplace (795). These last four attempts assumed implicitly Girard s assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. At the end of the 8th century, two new proofs were published which did not assume the existence of roots. One of them, due to James Wood and mainly algebraic, was published in 798 and it was totally ignored. Wood s proof had an algebraic gap. Another was published by Gauss, in his doctoral thesis in 799, and it was mainly geometric, but it had a topological gap. A rigorous proof was published by Jean Argand, a Swiss accountant, in 806; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 86 and another version of his original proof in The first textbook containing a proof of the theorem was Cauchy s Cours d analyse de l cole Royale Polytechnique (82). It contained Argand s proof, although Argand was not credited there (Fundamental theorem of algebra).