Complex Analysis: A Round-Up

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1 Complex Analysis: A Round-Up October 1, 2009 Sergey Lototsky, USC, Dept. of Math. *** 1

2 Prelude: Arnold s Principle Valdimir Igorevich Arnold (b. 1937): Russian The Arnold Principle. If a notion bears a personal name, then this name is not the name of the discoverer. The Berry Principle. The Arnold Principle is applicable to itself. (M. Berry was a student of Arnold) Two other quotations by V. I. Arnold: Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. ***************************************************************** Genuine mathematicians do not gang up, but the weak need gangs in order to survive. Sergey Lototsky, USC, Dept. of Math. *** 2

3 Complex Numbers: The Time Line First appearance: 50 A.D. (Heron of Alexandria) Necessity for cubic equations: 1545 (Cardano) The term imaginary : 1630 (Descartes) De Moivre s formula (cosθ + i sin θ) n = cosnθ + i sinnθ: 1722 The notation i: 1777 (Euler) Development of complex analysis: first half of 19th century Calculus done right: second half of 19th century Riemann s zeta function: 1859 The prime number theorem: 1895 Zeroes of the Riemann zeta function: currently, a million-dollar question Main reference: An Imaginary Tale: The Story of 1, by Paul J. Nahin, Princeton University Press, Sergey Lototsky, USC, Dept. of Math. *** 3

4 First Appearance Heron of Alexandria (c. 10 c. 70): Greek Area of triangle: p(p a)(p b)(p c), p = (a + b + c)/2; something similar for volumes. Things can become negative under the square root... Sergey Lototsky, USC, Dept. of Math. *** 4

5 Cardano s formula: x 3 = px + q; x = 3 q q p Cubic Equation q 2 q 2 4 p3 27 Example: x 3 = 15x + 4 Obviously x = 4, 2 ± 3 by Cardano s formula: x = Some history: first discovered around 1505 by Scipione del Ferro; learned by his student Fior and used to challenge Niccolo Tartaglia Fontana, who rediscovered the result the night before the challenge; Girolamo Cardano learned the result, but not the derivation, from Fontana, then re-derived and published it in 1545, in full generality and with full credits. Sergey Lototsky, USC, Dept. of Math. *** 5

6 Beyond the Cubic Niels Henrik Abel ( ): Norwegian; proved impossibility to solve 5-th order equations (1820). Abelian group Evariste Galois ( ): French; classified the equations that can be solved (1829). Galois Theory Sergey Lototsky, USC, Dept. of Math. *** 6

7 The Complex Plane: Argand s diagram René Descartes (Cartesius) ( ): French (1637); left John Wallis ( ): English (1685); middle. Caspar Wessel ( ): Norwegian; worked as a surveyor; wrote in Danish (1797) Johann Carl Friedrich Gauss ( ): German (1799); right. Jean-Robert Argand ( ): Swiss; was a bookstore manager in Paris (1806). Sergey Lototsky, USC, Dept. of Math. *** 7

8 Euler Formula (cosθ + i sin θ) n = cosnθ + i sin nθ Abraham de Moivre ( ): French (1722) Leonhard Euler ( ): Swiss; e iz = cos(z) + i sin(z): 1748; his notation i: Sergey Lototsky, USC, Dept. of Math. *** 8

9 Cauchy-Riemann Equations Augustin Louis Cauchy ( ): French Georg Friedrich Bernhard Riemann ( ): German (1851) Sergey Lototsky, USC, Dept. of Math. *** 9

10 The Integral Theorem of Cauchy Augustin Louis Cauchy ( ): French (1825: f continuous) Édouard Goursat ( ): French (1900: no continuity of f ) Giacinto Morera ( ): Italian (converse statement) Sergey Lototsky, USC, Dept. of Math. *** 10

11 Liouville s Theorem Joseph Liouville ( ): French 1 The first transcendental number (1844): = k! k=1 Also: Liouville s theorem in dynamical systems; Liouville s formula for second-order linear ODEs; Sturm-Liouville theory of two-point boundary-value problems. Sergey Lototsky, USC, Dept. of Math. *** 11

12 The Series Colin Maclaurin ( ): Scottish Brook Taylor ( ): English Pierre Alphonse Laurent ( ): French (1843) Sergey Lototsky, USC, Dept. of Math. *** 12

13 Conformal Mappings Georg Friedrich Bernhard Riemann ( ): German (1851: The Riemann mapping theorem) August Ferdinand Möbius ( ): German; Möbius transformation extends to higher dimensions Nikolai Yegorovich Zhukovsky ( ): Russian Sergey Lototsky, USC, Dept. of Math. *** 13

14 Ordinary Differential Equations Lazarus Immanuel Fuchs ( ): German Ferdinand Georg Frobenius ( ): German Charles Hermite ( ): French; proved that e is transcendental (1873) Sergey Lototsky, USC, Dept. of Math. *** 14

15 Making It Rigorous: Back to Real Numbers Georg Ferdinand Ludwig Philipp Cantor ( ): German; 1873: countability of rationals; uncountability of transcendentals Julius Wihelm Richard Dedekind ( ): German (1858: Dedekind cut) Karl Theodor Wilhelm Weierstrass ( ): German (1864) Sergey Lototsky, USC, Dept. of Math. *** 15

16 The Prime Number Theorem lim n number of primes n n/ ln n = 1 Jacques Salomon Hadamard ( ): French (1896) Charles Jean Gustave Nicolas Baron de la Valle Poussin ( ): Belgian (1896) Sergey Lototsky, USC, Dept. of Math. *** 16

17 The Most Famous Singular Point ζ(z) = k 1 1, Re(z) > 1. kz Can extend to a function analytic for all z EXCEPT z 0 = 1 This gives the Riemann zeta function. z 0 = 1 is a simple pole with Res equal to 1. An interesting equality: ( lim ζ(z) 1 ) ζ 1 z 1 = lim n ( n k=1 ) 1 k ln n = γ = (Euler s constant) Riemann hypothesis: ζ(z) = 0 if and only if Re(z) = 1 2. Sergey Lototsky, USC, Dept. of Math. *** 17

18 The Bottom Line The shortest path between two truths in the real domain passes through the complex domain J. Hadamard Sergey Lototsky, USC, Dept. of Math. *** 18

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