Complex Analysis: A Round-Up
|
|
- Adam Vincent Holmes
- 5 years ago
- Views:
Transcription
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
COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS
COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall
More informationChapter 1. Complex Numbers. 1.1 Complex Numbers. Did it come from the equation x = 0 (1.1)
Chapter 1 Complex Numbers 1.1 Complex Numbers Origin of Complex Numbers Did it come from the equation Where did the notion of complex numbers came from? x 2 + 1 = 0 (1.1) as i is defined today? No. Very
More informationax 2 +bx+c = 0, M = R 4 X
Complex Numbers Why shall we study complex analysis? We list several examples to illustrate why shall we study complex analysis. (Algebra) If we only limit ourselves to real numbers, the quadratic equation
More informationChapter 3.4: Complex Zeros of Polynomials
Chapter 3.4: Complex Zeros of Polynomials Imaginary numbers were first encountered in the first century in ancient Greece when Heron of Alexandria came across the square root of a negative number in his
More informationGalois theory - Wikipedia, the free encyclopedia
Page 1 of 8 Galois theory From Wikipedia, the free encyclopedia In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory
More informationMATH 5400, History of Mathematics
MATH 5400, History of Mathematics Lecture 10: 1900 Professor: Peter Gibson pcgibson@yorku.ca http://people.math.yorku.ca/pcgibson/math5400 February 16, 2017 In 1896 two mathematicians, working independently,
More informationLecture 2: What is Proof?
Lecture 2: What is Proof? Math 295 08/26/16 Webster Proof and Its History 8/2016 1 / 1 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct
More informationO1 History of Mathematics Lecture XIII Complex analysis. Monday 21st November 2016 (Week 7)
O1 History of Mathematics Lecture XIII Complex analysis Monday 21st November 2016 (Week 7) Summary Complex numbers Functions of a complex variable The Cauchy Riemann equations Contour integration Cauchy
More information524 partial differential equations
Index adjoint operator, 114 airfoils, 334 analog signal, 386 analytic function, 307 angular frequency, 372 area element, 217 Argand diagram, 287 associated Laguerre polynomials, 210 associated Legendre
More informationMath 404: History of Math. Stories, Teasers, and General Information
Math 404 History of Math: Stories,, and 4/19/2012 Italians Feud Over Cubics Scipione del Ferro Solved x 3 nx = m Italians Feud Over Cubics Scipione del Ferro Solved x 3 nx = m Antonio Fior Student under
More informationSmalltalk 9/26/13. Is it all in your imagination? Brian Heinold
Smalltalk 9/26/13 Is it all in your imagination? Brian Heinold What is i? Definition: i = 1 What is i? Definition: i = 1 Specifically, i is a number such that i 2 = 1. What is i? Definition: i = 1 Specifically,
More informationis the same, no matter in which order we multiply them. We can therefore unambiguously write a 1 a n. This product is also often denoted by n
Fields 2 The way of mathematical thought is twofold: the mathematician first proceeds inductively from the particular to the general and then deductively from the general to the particular. Moreover, throughout
More informationSinfonia. Professor Hong Guo 1
Sinfonia Professor Hong Guo (hongguo@pku.edu.cn) IQE@EE.EECS.PKU CREAM@IQE.EE.EECS.PKU 1 CREAM@IQE.EE.EECS.PKU 2 CREAM@IQE.EE.EECS.PKU 3 CREAM@IQE.EE.EECS.PKU 4 CREAM@IQE.EE.EECS.PKU 5 CREAM@IQE.EE.EECS.PKU
More informationComplex Analysis I MAST31006
Complex Analysis I MAST36 Lecturer: Ritva Hurri-Syrjänen Lectures: Tuesday: :5-2:, Wednesday: 4:5-6: University of Helsinki June 2, 28 Complex Analysis I CONTENTS Contents Background 4. The vector space
More informationLecture 13: Series Solutions near Singular Points
Lecture 13: Series Solutions near Singular Points March 28, 2007 Here we consider solutions to second-order ODE s using series when the coefficients are not necessarily analytic. A first-order analogy
More informationA Brief History of Algebra
A Brief History of Algebra The Greeks: Euclid, Pythagora, Archimedes Indian and arab mathematicians Italian mathematics in the Renaissance The Fundamental Theorem of Algebra Hilbert s problems 1 Pythagoras,
More informationO1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis. Monday 30th October 2017 (Week 4)
O1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis Monday 30th October 2017 (Week 4) Summary French institutions Fourier series Early-19th-century rigour Limits, continuity,
More informationMATH3500 The 6th Millennium Prize Problem. The 6th Millennium Prize Problem
MATH3500 The 6th Millennium Prize Problem RH Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime
More informationO1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis. Monday 31st October 2016 (Week 4)
O1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis Monday 31st October 2016 (Week 4) Summary French institutions Fourier series Early-19th-century rigour Limits, continuity,
More informationComplex Numbers, Basics
i Complex Numbers, Basics The shortest path between two truths in the real domain passes through the complex domain." (Jaques Hadamard 1865 1963) 1 E1 1 E2 Introduction It was one of the early problems
More informationFourier series: Fourier, Dirichlet, Poisson, Sturm, Liouville
Fourier series: Fourier, Dirichlet, Poisson, Sturm, Liouville Joseph Fourier (1768-1830) upon returning from Egypt in 1801 was appointed by Napoleon Prefect of the Department of Isères (where Grenoble
More informationSets and Infinity. James Emery. Edited: 2/25/ Cardinal Numbers 1. 2 Ordinal Numbers 6. 3 The Peano Postulates for the Natural Numbers 7
Sets and Infinity James Emery Edited: 2/25/2017 Contents 1 Cardinal Numbers 1 2 Ordinal Numbers 6 3 The Peano Postulates for the Natural Numbers 7 4 Metric Spaces 8 5 Complete Metric Spaces 8 6 The Real
More informationO1 History of Mathematics Lecture XII 19th-century rigour in real analysis, part 2: real numbers and sets. Monday 14th November 2016 (Week 6)
O1 History of Mathematics Lecture XII 19th-century rigour in real analysis, part 2: real numbers and sets Monday 14th November 2016 (Week 6) Summary Proofs of the Intermediate Value Theorem revisited Convergence
More information[1] Robert G. Bartle, The Elements of Real Analysis. Second Edition. John Wiley and Sons, New York, 1964.
Bibliography [1] Robert G. Bartle, The Elements of Real Analysis. Second Edition. John Wiley and Sons, New York, 1964. [2] Robert G. Bartle, Return to the Riemann Integral. American Mathematical Monthly,
More information1. Complex Numbers. John Douglas Moore. July 1, 2011
1. Complex Numbers John Douglas Moore July 1, 2011 These notes are intended to supplement the text, Fundamentals of complex analysis, by Saff and Snider [5]. Other often-used references for the theory
More informationMthEd/Math 300 Williams Fall 2011 Midterm Exam 3
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 3 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More informationO1 History of Mathematics Lecture XII 19th-century rigour in real analysis, part 2: real numbers and sets. Monday 13th November 2017 (Week 6)
O1 History of Mathematics Lecture XII 19th-century rigour in real analysis, part 2: real numbers and sets Monday 13th November 2017 (Week 6) Summary Proofs of the Intermediate Value Theorem revisited Convergence
More informationBeyond Newton and Leibniz: The Making of Modern Calculus. Anthony V. Piccolino, Ed. D. Palm Beach State College Palm Beach Gardens, Florida
Beyond Newton and Leibniz: The Making of Modern Calculus Anthony V. Piccolino, Ed. D. Palm Beach State College Palm Beach Gardens, Florida Calculus Before Newton & Leibniz Four Major Scientific Problems
More informationFour Mathema,cians Who Shaped Our Understanding of Calculus
Four Mathema,cians Who Shaped Our Understanding of Calculus David Bressoud St. Paul, MN WisMATYC Peewaukee, WI September 28, 2013 PowerPoint available at www.macalester.edu/~bressoud/talks The task of
More informationChapter 2 Lambert Summability and the Prime Number Theorem
Chapter 2 Lambert Summability and the Prime Number Theorem 2. Introduction The prime number theorem (PNT) was stated as conjecture by German mathematician Carl Friedrich Gauss (777 855) in the year 792
More informationStories from the Development Of Real Analysis
Stories from the Development Of Real Analysis David Bressoud Macalester College St. Paul, MN PowerPoint available at www.macalester.edu/~bressoud/talks Seaway Sec(on Mee(ng Hamilton College Clinton, NY
More information2.3. Polynomial Equations. How are roots, x-intercepts, and zeros related?
.3 Polynomial Equations Suppose the volume, V, in cubic centimetres, of a block of ice that a sculptor uses to carve the wings of a dragon can be modelled by V(x) 9x 3 60x 49x, where x represents the thickness
More informationIndex. c, 400 Ørstead, Christian, 397 matrix:symmetric, 160
Index c, 400 Ørstead, Christian, 397 matrix:symmetric, 160 adjoint operator, 243 Ampère,André-Marie, 397 amplitude, 55 analog signal, 343 analytic function, 288 angular frequency, 55, 332 anticommutative,
More informationIndex. Excerpt from "Intermediate Algebra" 2014 AoPS Inc. Copyrighted Material INDEX
Index [x], 529, 572 \, 29 \, 29 [, 28 1, 3 dxe, 529 539 bxc, 529 539, 201 Q, 313 331 P, 313 331 {a i }, sequence notation, 313 {x}, fractional part, 529 539 Abel Prize, 507 Abel Summation, 368 Abel, Niels,
More informationAbstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley
Abstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley Janet Heine arnett 12 January 2010 Introduction The problem of solving polynomial equations is nearly as
More informationWhat do you think are the qualities of a good theorem? it solves an open problem (Name one..? )
What do you think are the qualities of a good theorem? Aspects of "good" theorems: short surprising elegant proof applied widely: it solves an open problem (Name one..? ) creates a new field might be easy
More informationContents Part A Number Theory Highlights in the History of Number Theory: 1700 BC 2008
Contents Part A Number Theory 1 Highlights in the History of Number Theory: 1700 BC 2008... 3 1.1 Early Roots to Fermat... 3 1.2 Fermat... 6 1.2.1 Fermat s Little Theorem... 7 1.2.2 Sums of Two Squares...
More informationAn Introduction to Analysis on the Real Line for Classes Using Inquiry Based Learning
An Introduction to Analysis on the Real Line for Classes Using Inquiry Based Learning Helmut Knaust Department of Mathematical Sciences The University of Texas at El Paso El Paso TX 79968-0514 hknaust@utep.edu
More informationAbstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley
Abstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley Janet Heine arnett janet.barnett@colostate-pueblo.edu Department of Mathematics and Physics Colorado State
More informationFAMOUS SCIENTISTS: LC CHEMISTRY
FAMOUS SCIENTISTS: LC CHEMISTRY Study online at quizlet.com/_6j280 1. SVANTE AUGUST ARRHENIUS 4. ANTOINE HENRI BECQUEREL He developed a theory of acids and bases on how they form ions in solution. He also
More information36 CHAPTER 2. COMPLEX-VALUED FUNCTIONS. In this case, we denote lim z z0 f(z) = α.
36 CHAPTER 2. COMPLEX-VALUED FUNCTIONS In this case, we denote lim z z0 f(z) = α. A complex-valued function f defined in A is called continuous at z 0 A if lim z z 0 f(z) = f(z 0 ). Theorem 2.1.1 Let A
More informationMath 495 Dr. Rick Kreminski Colorado State University Pueblo November 19, 2014 The Riemann Hypothesis: The #1 Problem in Mathematics
Math 495 Dr. Rick Kreminski Colorado State University Pueblo November 19, 14 The Riemann Hypothesis: The #1 Problem in Mathematics Georg Friedrich Bernhard Riemann 1826-1866 One of 6 Million Dollar prize
More informationReview Sheet for the Final Exam of MATH Fall 2009
Review Sheet for the Final Exam of MATH 1600 - Fall 2009 All of Chapter 1. 1. Sets and Proofs Elements and subsets of a set. The notion of implication and the way you can use it to build a proof. Logical
More informationThe Different Sizes of Infinity
The Different Sizes of Infinity New York City College of Technology Cesar J. Rodriguez November 11, 2010 A Thought to Ponder At... Does Infinity Come in Varying Sizes? 2 Points of Marked Interest(s) General
More informationCHAPTER 5. Fields. 47 Historical Introduction
CHAPTER 5 Fields 47 Historical Introduction For a long time in the history of mathematics, algebra was understood to be the study of roots of polynomials. This must be clearly distinguished from numerical
More informationIn Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-1: Basic Ideas 1 Introduction
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationIntroduction to Complex Numbers Complex Numbers
Introduction to SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/ Retell, Activating Prior Knowledge, Create Representations The equation x 2 + 1 = 0 has special historical and mathematical significance.
More informationComplex Numbers: A Brief Introduction. By: Neal Dempsey. History of Mathematics. Prof. Jennifer McCarthy. July 16, 2010
1 Complex Numbers: A Brief Introduction. By: Neal Dempsey History of Mathematics Prof. Jennifer McCarthy July 16, 2010 2 Abstract Complex numbers, although confusing at times, are one of the most elegant
More informationNeedles and Numbers. The Buffon Needle Experiment
eedles and umbers This excursion into analytic number theory is intended to complement the approach of our textbook, which emphasizes the algebraic theory of numbers. At some points, our presentation lacks
More informationSection V.9. Radical Extensions
V.9. Radical Extensions 1 Section V.9. Radical Extensions Note. In this section (and the associated appendix) we resolve the most famous problem from classical algebra using the techniques of modern algebra
More informationINDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43
INDEX Abel s identity, 131 Abel s test, 131 132 Abel s theorem, 463 464 absolute convergence, 113 114 implication of conditional convergence, 114 absolute value, 7 reverse triangle inequality, 9 triangle
More informationThe Princeton Companion to Mathematics
The Princeton Companion to Mathematics EDITOR Timothy Gowers University of Cambridge ASSOCIATE EDITORS June Barrow-Green The Open University Imre Leader University of Cambridge Princeton University Press
More informationModern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS
Modern Analysis Series Edited by Chung-Chun Yang AN INTRODUCTION TO COMPLEX ANALYSIS Classical and Modern Approaches Wolfgang Tutschke Harkrishan L. Vasudeva ««CHAPMAN & HALL/CRC A CRC Press Company Boca
More informationApplication of Series in Heat Transfer - transient heat conduction
Application of Series in Heat Transfer - transient heat conduction By Alain Kassab Mechanical, Materials and Aerospace Engineering UCF EXCEL Applications of Calculus Part I background and review of series
More information11-6. Solving All Polynomial Equations. Vocabulary. What Types of Numbers Are Needed to Solve Polynomial Equations? Lesson
Chapter 11 Lesson 11-6 Solving All Polynomial Equations Vocabulary double root, root of multiplicity 2 multiplicity BIG IDEA Every polynomial equation of degree n 1 has exactly n solutions, counting multiplicities.
More informationComplex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number
Complex Numbers December, 206 Introduction 2 Imaginary Number In your study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions. For example, try as
More informationAppendix 1. Cardano s Method
Appendix 1 Cardano s Method A1.1. Introduction This appendix gives the mathematical method to solve the roots of a polynomial of degree three, called a cubic equation. Some results in this section can
More informationChapter 3: Complex Numbers
Chapter 3: Complex Numbers Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 3: Complex Numbers Semester 1 2018 1 / 48 Philosophical discussion about numbers Q In what sense is 1 a number? DISCUSS
More informationThe Value of Imaginary Numbers. Most of us are familiar with the numbers that exist on a one-dimensional scale called the number line, as
Paige Girardi Girardi 1 Professor Yolande Petersen Math 101, MW 11:40-1:05 April 6 2016 The Value of Imaginary Numbers Most of us are familiar with the numbers that exist on a one-dimensional scale called
More informationThe Prime Number Theorem
Chapter 3 The Prime Number Theorem This chapter gives without proof the two basic results of analytic number theory. 3.1 The Theorem Recall that if f(x), g(x) are two real-valued functions, we write to
More informationLet π and e be trancendental numbers and consider the case:
Jonathan Henderson Abstract: The proposed question, Is π + e an irrational number is a pressing point in modern mathematics. With the first definition of transcendental numbers coming in the 1700 s there
More informationThe Roots of Early Group Theory in the Works of Lagrange
Ursinus College Digital Commons @ Ursinus College Abstract Algebra Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Fall 2017 The Roots of Early Group Theory
More informationThe group law on elliptic curves
Mathematisch Instituut Universiteit Leiden Elliptic curves The theory of elliptic curves is a showpiece of modern mathematics. Elliptic curves play a key role both in the proof of Fermat s Last Theorem
More informationINTRODUCTION TO TRANSCENDENTAL NUMBERS
INTRODUCTION TO TRANSCENDENTAL NUBERS VO THANH HUAN Abstract. The study of transcendental numbers has developed into an enriching theory and constitutes an important part of mathematics. This report aims
More information1230, notes 16. Karl Theodor Wilhelm Weierstrass, November 18, / 18
1230, notes 16 Karl Theodor Wilhelm Weierstrass, 1815-1897 November 18, 2014 1 / 18 1230, notes 16 Karl Theodor Wilhelm Weierstrass, 1815-1897 Left university without a degree (ignored what he was supposed
More informationRigorization of Calculus. 18 th Century Approaches,Cauchy, Weirstrass,
Rigorization of Calculus 18 th Century Approaches,Cauchy, Weirstrass, Basic Problem In finding derivatives, pretty much everyone did something equivalent to finding the difference ratio and letting. Of
More informationEuler s Equation in Complex Analysis
Euler s Equation in Complex Analysis Leqi Wang July 2017 Math 190s Duke University!1 Euler s Equation in Complex Analysis Abstract Euler s equation is one of the most beautiful identities throughout the
More information11 th Annual Harvard-MIT Mathematics Tournament
11 th Annual Harvard-MIT Mathematics Tournament Saturday 23 February 2008 Guts Round.. 1. [5] Determine all pairs (a, b) of real numbers such that 10, a, b, ab is an arithmetic progression. 2. [5] Given
More informationSyllabus: for Complex variables
EE-2020, Spring 2009 p. 1/42 Syllabus: for omplex variables 1. Midterm, (4/27). 2. Introduction to Numerical PDE (4/30): [Ref.num]. 3. omplex variables: [Textbook]h.13-h.18. omplex numbers and functions,
More informationThe arithmetic geometric mean (Agm)
The arithmetic geometric mean () Pictures by David Lehavi This is part of exercise 5 question 4 in myc> calculus class Fall 200: a = c a n+ =5a n 2 lim n an = a = a an + bn a n+ = 2 b = b b n+ = a nb n
More informationSymmetry Anyone? Willy Hereman. After Dinner Talk SANUM 2008 Conference Thursday, March 27, 2007, 9:00p.m.
Symmetry Anyone? Willy Hereman After Dinner Talk SANUM 2008 Conference Thursday, March 27, 2007, 9:00pm Tribute to Organizers Karin Hunter Karin Hunter Andre Weideman Karin Hunter Andre Weideman Ben Herbst
More informationAlgebra and Geometry in the Sixteenth and Seventeenth Centuries
Algebra and Geometry in the Sixteenth and Seventeenth Centuries Introduction After outlining the state of algebra and geometry at the beginning of the sixteenth century, we move to discuss the advances
More informationPolynomials and Taylor s Approximations
214 TUTA/IOE/PCU Journal of the Institute of Engineering, 2016, 12(1): 214-221 TUTA/IOE/PCU Printed in Nepal Polynomials and Taylor s Approximations Nhuchhe Shova Tuladhar Department of Science and Humanities,
More informationCardano and the Solution of the Cubic. Bryan Dorsey, Kerry-Lyn Downie, and Marcus Huber
Cardano and the Solution of the Cubic Bryan Dorsey, Kerry-Lyn Downie, and Marcus Huber Pacioli In 1494, the Italian Luca Pacioli produced his volume titled Summa de Arithmetica. In this a step was made
More informationThe Complex Numbers c ). (1.1)
The Complex Numbers In this chapter, we will study the basic properties of the field of complex numbers. We will begin with a brief historic sketch of how the study of complex numbers came to be and then
More informationGALOIS THEORY FOR PHYSICISTS
GALOIS THEORY FOR PHYSICISTS Spontaneous Symmetry Breaking and the Solution to the Quintic Tatsu Takeuchi @Kyoto Sangyo University, Maskawa-Juku January 12, 2012 DISCLAIMER: Start with the complex number
More informationPart IB. Complex Analysis. Year
Part IB Complex Analysis Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section I 2A Complex Analysis or Complex Methods 7 (a) Show that w = log(z) is a conformal
More informationComplex Varables Lecture Notes for Math 122A. John Douglas Moore
Complex Varables Lecture Notes for Math 122A John Douglas Moore July 27, 2011 Contents 1 Complex Numbers 1 1.1 Field axioms............................. 2 1.2 Complex numbers..........................
More informationQualifying Exam Complex Analysis (Math 530) January 2019
Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,
More informationVII.8. The Riemann Zeta Function.
VII.8. The Riemann Zeta Function VII.8. The Riemann Zeta Function. Note. In this section, we define the Riemann zeta function and discuss its history. We relate this meromorphic function with a simple
More informationRiemann s Zeta Function and the Prime Number Theorem
Riemann s Zeta Function and the Prime Number Theorem Dan Nichols nichols@math.umass.edu University of Massachusetts Dec. 7, 2016 Let s begin with the Basel problem, first posed in 1644 by Mengoli. Find
More informationMathematical Transition
Lecture 3 Mathematical Transition For the construction of the regular pentagon, we used the five solutions, z 0, z 1, z 2, z 3, z 4,of Z 5 1=0, thus the five numbers z k =cos(2πk/5) + i sin(2πk/5), k =0,
More informationAH Complex Numbers.notebook October 12, 2016
Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does
More informationASSIGNMENT-1 M.Sc. (Previous) DEGREE EXAMINATION, DEC First Year MATHEMATICS. Algebra. MAXIMUM MARKS:30 Answer ALL Questions
(DM1) ASSIGNMENT-1 Algebra MAXIMUM MARKS:3 Q1) a) If G is an abelian group of order o(g) and p is a prime number such that p α / o(g), p α+ 1 / o(g) then prove that G has a subgroup of order p α. b) State
More informationFirst order Friedmann equation = t f. and α. Substitute ã α = α cos θ = t f = α(θ f sin θ f ) ã f = α(1 cos θ f ) = α(1 cos θ)
Alan Guth, Introduction to Non-Euclidean Spaces (After finishing dynamics of homogeneous expansion), Lecture 10, October 10, 2013, p. 1. Summary of Lecture 9 First order Friedmann equation = t f ãf ãdã
More informationAn improper integral and an infinite series
An improper integral and an infinite series A. Baltimore one of the old cities in the United States Yue Kwok Choy In summer of 2010, I had great time visiting my daughter, Wendy, who is living in a small
More informationmass vs. weight. 392 dependent variable, 2 derivative(s) of a power series. 459 Descartes, René, 201 Devil s curve, 126 Difference Law of limits, 36 D
# #, # definition of a limit, 31,64 A absolute maximum and minimum, 199 absolute maximum and minimum values, 199 absolute value function, 5 absolutely convergent series, 446 acceleration, 282 Airy function,
More informationLecture IX. Definition 1 A non-singular Sturm 1 -Liouville 2 problem consists of a second order linear differential equation of the form.
Lecture IX Abstract When solving PDEs it is often necessary to represent the solution in terms of a series of orthogonal functions. One way to obtain an orthogonal family of functions is by solving a particular
More informationA history of Topology
A history of Topology Version for printing Geometry and topology index History Topics Index Topological ideas are present in almost all areas of today's mathematics. The subject of topology itself consists
More informationIntroduction on Bernoulli s numbers
Introduction on Bernoulli s numbers Pascal Sebah and Xavier Gourdon numbers.computation.free.fr/constants/constants.html June 2, 2002 Abstract This essay is a general and elementary overview of some of
More informationMATH DAY 2018 TEAM COMPETITION
An excursion through mathematics and its history (the middle ages and beyond)(and some trivia) MATH DAY 2018 TEAM COMPETITION Made possible by A quick review of the rules History (or trivia) questions
More informationElementary Number Theory
Elementary Number Theory 21.8.2013 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What
More informationO1 History of Mathematics Lecture X The 19th-century beginnings of modern algebra. Monday 7th November 2016 (Week 5)
O1 History of Mathematics Lecture X The 19th-century beginnings of modern algebra Monday 7th November 2016 (Week 5) Summary Lagrange s ideas (1770/71) Cauchy and substitutions (1815) Classical age of theory
More informationPMATH 300s P U R E M A T H E M A T I C S. Notes
P U R E M A T H E M A T I C S Notes 1. In some areas, the Department of Pure Mathematics offers two distinct streams of courses, one for students in a Pure Mathematics major plan, and another for students
More informationCOMPLEX ANALYSIS. Rudi Weikard
COMPLEX ANALYSIS Lecture notes for MA 648 Rudi Weikard 3 2 2 1 1 0 0-2 -1-1 0 1 2-2 Version of December 8, 2016 Contents Preface 1 Chapter 1. The complex numbers: algebra, geometry, and topology 3 1.1.
More informationContents. 1 Solving algebraic equations Field extensions Polynomials and irreducibility Algebraic extensions...
Contents 1 Solving algebraic equations............................................ 1 2 Field extensions....................................................... 9 3 Polynomials and irreducibility.........................................
More informationComplex Variables. Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit.
Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit. 1. The TI-89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim
More informationSyllabuses for Honor Courses. Algebra I & II
Syllabuses for Honor Courses Algebra I & II Algebra is a fundamental part of the language of mathematics. Algebraic methods are used in all areas of mathematics. We will fully develop all the key concepts.
More information