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Transcript
1 A Brief History of √ −1 and Complex Analysis • The quadratic formula has been known since Babylonian times, but negative square roots were seen as coming from impossible solutions and were discarded. • The depressed cubic equation (2nd order term removed) was first solved by Scipione del Ferro (1465–1526) who kept it secret. (University positions could be won and lost through mathematical challenges.) His solution to x3 + px = q (both p and q positive) is equivalent to � � � � 3 2 3 q 3 q p p3 q q2 + + + − + x= 2 4 27 2 4 27 Cubics were all reduced to one of three cases without negative coefficients: x3 + px = q, √ x3 = px + q, and x3 + q = px. Everyone still viewed −1 as an impossible quantity. • The depressed cubic was rediscovered by Tartaglia (Niccolo Fontana 1500–1577) in 1535. • Girolamo Cardano (1571–1576) learned the formula from Tartaglia under an oath of secrecy. Later, Cardan was shown a book written by del Ferro and saw that as releasing his oath. He published a method in Ars Magna giving credit to both del Ferro and Tartaglia. Cardan extended the solution to all cubics x3 +a1 x2 +a2 x+a3 = 0 via the transformation x = X−a1 /3. He called equations with complex roots sophistic, but used them formally to obtain real roots using the identity z + z̄ = 2Re(z). • Rafael Bombelli (1526–1572) wrote of a “wild idea” in his Algebra of 1542: Formal manipu√ lation of −1 to obtain real results. • François Viète (1540–1603) used cosine and arccosine to avoid imaginary quantities. • Leonhard Euler (1707–1783) wrote to Johann Bernoulli (his former teacher) in 1740 the result now known as “Euler’s Formula” √ √ eθ −1 = cos(θ) + sin(θ) −1 He published the formula in 1748 in his Introductio Analysis Infinitorum. The result had been published first by Roger Cotes (1682–1716) in 1714, but in logarithmic form. Cotes writing was so obscure that no one understood his work. • Abraham De Moivre (1667–1754) never published “his” theorem √ �1/m √ � = cos (θ/m) + sin (θ/m) −1 cos(θ) + sin(θ) −1 but, Isaac Newton and Leonhard Euler (1748) attributed it to him • Jean d’Alembert (1717–1783) presented a proof of the Fundamental Theorem of Algebra using complex values. His proof was flawed, but led Euler, Gauss, and others to work on the result. • Euler invented the ι notation in 1777. Euler wrote in his Algebra in 1770 √ √ All such expressions as −1, −2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers, we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible. 1 Euler also gave a “proof” that √ −2 · √ −3 = √ 6. • Caspar Wessel (1745–1818) invented the geometric interpretation as x + ιy in xy-plane in 1797. His work had no impact outside Denmark until his paper was rediscovered in 1895. Wessel presented complex multiplication geometrically as the product of magnitudes and sum of arguments: To multiply by ι is a counter-clockwise rotation by 90 degrees. • Jean-Robert Argand (1768–1822) rediscovered Wessel’s idea independently in 1806. • Adrien-Quentin Buée (1748–1826) also rediscovered Wessel’s idea in 1806. He stated √ “ −1 is the sign of perpendicularity.” • Augustin-Louis Cauchy (1789–1857) investigated functions of a complex variable. He developed the calculus of complex functions (1814), discovered the connection between derivative and integral, investigated infinite series, and defined analytic functions. • Karl Freidrich Gauss (1777–1855) introduced the term complex number. He proved the Fundamental Theorem of Algebra and in 1849 extended the result to complex polynomials. • William Rowan Hamilton (1806–1865) defined complex numbers as ordered pairs such that (a, b) + (c, d) = (a + b, c + d) and (a, b)(c, d) = (ac − bd, ad + bc) in 1835. Hamilton wanted a purely algebraic definition of complex numbers. • Karl von Weierstrass (1815–1897) proved (1863) that the complex numbers are the only commutative algebraic extension of the real numbers. Gauss had promised, but failed, to give a proof of this theorem. • Victor Puiseux (1820–1883) defined branches and was the first to distinguish poles, essential points and branch points. • Georg Fredreich Bernhard Riemann’s (1826–1866) doctoral thesis (1851), directed by Gauss, studied the theory of complex variables and what we now call Riemann surfaces, introducing topological methods into complex function theory. 2 2 People of Complex Analysis Scipione del Ferro Nicollo Tartaglia Gerolamo Cardan François Viète Leonhard Euler Johann Bernoulli Roger Cotes Auguste De Moivre Issac Newton A. D’Alembert Augustin Cauchy Karl Gauss William Hamilton Karl Weiertrass Victor Puiseux Georg Riemann 3 3 Early Texts with √ −1 The images below show the frontispieces of the texts. Girolamo Cardano Rafael Bombelli 4