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 s theorems and theory What is an analytic function?
Early ideas about complex numbers Before 1600, very faint beginnings: Cardano (1545) [from quadratics] Bombelli (1572) [from cubics] Harriot (c. 1600) [from quartics] But: For the most part such roots were ignored: negative roots were described merely as false, but complex roots as impossible. (Mathematics emerging, p. 459.)
Ideas about complex numbers in the 17th century John Wallis, A treatise of algebra (1685): complex numbers based on insights derived from Euclidean geometry trigonometry properties of conics (See: Mathematics emerging, 15.1.1.)
Wallis: justification of imaginary numbers A man starts at A and walks 5 yds to B, then retreats 2 yds to C: overall, he has covered 3 yds. If he instead retreats 8 yds to D, then we may say that he has covered -3 yds. Somewhere on the seashore, we gain 26 units of land from the sea, but lose 10 units. Thus, we have gained 16 units overall; if this is a perfect square, then it has side 4 units of length. If instead we lose 26 units of land, but gain 10, then we have lost 16 units overall, or gained -16.The area in question (assumed to be a square) might therefore be viewed as having side 16. (see: Leo Corry, A brief history of numbers, OUP, 2015, pp. 184 185)
Wallis: imaginary numbers as geometric means (see: Leo Corry, A brief history of numbers, OUP, 2015, pp. 185 186)
A new Impossibility in Algebra John Wallis, A treatise of algebra, p. 267 Of negative squares :... requires a new Impossibility in Algebra.
Complex numbers in the 18th century (1) Johann Bernoulli used them freely: As early as 1702 Johann Bernoulli... included some useful transformations which converted real integrals to imaginary ones, or conversely. At this point neither he nor anyone else had any compunction about swapping between real and imaginary in this way, especially since such transformations were extremely helpful. (Mathematics emerging, p. 468.)
Johann Bernoulli, 1702 (1) Johann Bernoulli, Solution d un problème concernant le calcul intégrale,..., Mémoires de l Académie royale des sciences, 1702:
Johann Bernoulli, 1702 (2) Johann Bernoulli, Solution d un problème concernant le calcul intégrale,..., Mémoires de l Académie royale des sciences, 1702: In the same way the differential adz bb + zz will be transformed into the differential of an imaginary logarithm adt 2bt 1; and conversely. (See: Mathematics emerging, 15.2.1.)
Complex numbers in the 18th century (2) Leonhard Euler also used them freely: e.g., in Introductio in analysin infinitorum, 1748, 138: e +v 1 = cos.v + 1. sin.v & e v 1 = cos.v 1. sin.v (See: Mathematics emerging, 9.2.3.)
Complex numbers become familiar objects Carl Friedrich Gauss and his proofs of the Fundamental Theorem of Algebra, 1799, 1815, etc. Gauss also in his Disquisitiones arithmeticae (1801): arithmetic of quadratic forms such as x 2 + y 2 ; roots of unity and applications to cyclotomy; etc. Jean Robert Argand and his diagram, 1806 (or Caspar Wessel, 1797) Abel, Galois, Cauchy, Hamilton everyone.
Complex analysis The origins of complex analysis may be seen in early achievements by Johann Bernoulli, Euler, and others, using complex transformations to evaluate real integrals. But is substitution of complex variables for real variables permissible? Euler (posthumous, 1794): yes Laplace (1785, 1812): yes Poisson (1812): doubtful Cauchy (1814): inspired by Laplace, set to work on the problem
Sources for the origins of complex analysis Secondary: Katz: 17.3 (3rd ed.); 22.3 (brief ed.) Frank Smithies: Cauchy and the creation of complex function theory, Cambridge University Press, 1997 Primary: as quoted by Smithies; some extracts reproduced in Mathematics emerging, 15.2.
Real and complex analysis united
Cauchy as creator of complex analysis Some of Cauchy s contributions to complex analysis: integration along paths and contours (1814) [1827] calculus of residues (1826) integration formulae (1831) inferences about Taylor series expansions applications to evaluation of difficult definite integrals of real functions
The Cauchy Riemann equations If f (z) = f (x + iy) = u(x, y) + iv(x, y) and f (z) exists then u x = v y, u y = v x. Smithies traces the equations back to d Alembert 1752, then Euler 1757, d Alembert 1761, Euler 1775, Lagrange 1781,... Cauchy 1814 1831, Riemann 1851.
Contour integration Definition of integral b a f (z) dz : Gauss, 1811 (unpublished and uninfluential); Cauchy, 1827 (written 1814) [Cauchy 1818 1831] similar to 1823 definition of definite integral of a real function of a real variable [see Lecture X] Integral b a Gauss as above; f (z) dz independent of path from a to b : Cauchy as above.
Cauchy s integral formulae f (a) = 1 f (z) 2πi γ z a dz, f (n) (a) = n! f (z) dz 2πi γ (z a) n+1 (Cauchy in various versions from 1822 to 1831)
Taylor s Theorem for complex analytic functions Derivation of Taylor series in the form f (z) = c 0 + c 1 z + c 2 z 2 + + c n z n +, where c n = 1 f (z) 2πi γ (z a) n+1 dz = f (n) (a) n! from Cauchy s Integral Formula: done by Cauchy in 1831.
Cauchy s calculus of residues Recall: if f (z) = c n (z a) n, then Res (f ; a) = c 1. n= Cauchy, 1826: residues of simple functions; also beginnings of calculus of residues Cauchy, 1826 1831: f (z) dz = 2πi residues inside Γ Γ
Cauchy s motivation Applications to evaluation of definite integrals on R or [0, ). Example: cos x 1 + x 2 dx = π e [1814/1827] Example: 0 sin x x dx = π 2 [1814/1827] Example: 0 sin x cos x π dx = dx = x 0 x 2 [1814/1827] Evaluation of infinite series, inversion of Laplace transforms, etc., etc.
Conformality Recall: a map f : R 2 R 2 or C C is said to be conformal if it preserves angles. Suppose f (a) 0. Write f (a) = re iα. By definition ( ) f (a + h) = f (a) + h f (a) + η where η 0 as h 0. So f (a + h) f (a) = re iα h + ηh and the local effect of f is to move a to f (a), scale by r, and rotate by α. So f : C C is conformal at all points where f (z) 0. Katz traces this back to Euler and Gauss. It is explicit and fundamental in Riemann, 1851.
The word analytic Words analysis, analytic have had many meanings: Classical: a method of investigating a problem, the opposite of synthesis c. 1600: algebra became known as the analytic art or just analysis, using finite equations 1669: Newton introduced analysis with infinite equations, that is, infinite series 1748: Euler wrote on the analysis of infinitely large and infinitely small quantities 1790 1840: in sections of journals, the Académie des Sciences, etc., Analyse could mean pure mathematics though with a bias to algebra, calculus, etc.; compare Géométrie also meaning pure mathematics, but with (perhaps) spatial bias 1821: Cauchy s cours d analyse shows similarities with our analysis courses today
What is an analytic function? Lagrange, 1797: function is analytic if it has a power-series expansion Cauchy s point of departure, 1814 1831: treated complex functions that are continuous and satisfy the Cauchy Riemann equations (always true for analytic functions in the sense of Lagrange), but used no special terminology Weierstrass, 1860s and later; Riemann, 1867: complex function f f (z + h) f (z) is analytic if lim h 0 exists (in region of interest) h Oxford, 2016: we follow Weierstrass, Riemann using words holomorphic, meromorphic, etc. as variants of analytic, with slightly different meanings