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Contents Complex numbers Diophantus Italian rennaissance mathematicians Rene Descartes Abraham de Moivre Leonhard Euler Caspar Wessel Jean-Robert Argand Carl Friedrich Gauss

Contents (cont.) Complex functions Augustin Louis Cauchy Georg F. B. Riemann Cauchy – Riemann equation The use of complex numbers today Discussion???

Diophantus of Alexandria Circa 200/214 - circa 284/298 An ancient Greek mathematician He lived in Alexandria Diophantine equations Diophantus was probably a Hellenized Babylonian.

Area and perimeter problems Collection of taxes Right angled triangle Perimeter = 12 units Area = 7 square units ?

Can you find such a triangle? The hypotenuse must be (after some calculations) 29/6 units Then the other sides must have sum = 43/6, and product like 14 square units. You can’t find such numbers!!!!!

Italian rennaissance mathematicians They put the quadric equations into three groups (they didn’t know the number 0): ax² + b x = c ax² = b x + c ax² + c = bx

Italian rennaissance mathematicians Del Ferro (1465 – 1526) Found sollutions to: x³ + bx = c Antonio Fior Not that smart – but ambitious Tartaglia (1499 - 1557) Re-discovered the method – defeated Fior Gerolamo Cardano (1501 – 1576) Managed to solve all kinds of cubic equations+ equations of degree four. Ferrari Defeated Tartaglia in 1548

Cardano’s formula

Rafael Bombelli Made translations of Diophantus’ books Calculated with negative numbers Rules for addition, subtraction and multiplication of complex numbers

A classical example using Cardano’s formula Lets try to put in the number 4 for x 64 – 60 – 4 = 0 We see that 4 has to be the root (the positive root)

(Cont.) Cardano’s formula gives: Bombelli found that: WHY????

(Cont.)

Rene Descartes (1596 – 1650) Cartesian coordinate system a + ib i is the imaginary unit i² = -1

Abraham de Moivre (1667 - 1754) (cosx + i sinx)^n = cos(nx) + i sin(nx) z^n= 1 Newton knew this formula in 1676 Poor – earned money playing chess

Leonhard Euler 1707 - 1783 Swiss mathematician Collected works fills 75 volumes Completely blind the last 17 years of his life

Euler's formula in complex analysis

Caspar Wessel (1745 – 1818) The sixth of fourteen children Studied in Copenhagen for a law degree Caspar Wessel's elder brother, Johan Herman Wessel was a major name in Norwegian and Danish literature Related to Peter Wessel Tordenskiold

Wessels work as a surveyor Assistant to his brother Ole Christopher Employed by the Royal Danish Academy Innovator in finding new methods and techniques Continued study for his law degree Achieved it 15 years later Finished the triangulation of Denmark in 1796

Om directionens analytiske betegning On the analytic representation of direction Published in 1799 First to be written by a non-member of the RDA Geometrical interpretation of complex numbers Re – discovered by Juel in 1895 !!!!! Norwegian mathematicians (UiO) will rename the Argand diagram the Wessel diagram

Wessel diagram / plane

Om directionens analytiske betegning Vector addition

Om directionens analytiske betegning Vector multiplication An example:

(Cont.) The modulus is: The argument is : Then (by Wessels discovery):

Jean-Robert Argand (1768-1822) Non – professional mathematician Published the idea of geometrical interpretation of complex numbers in 1806 Complex numbers as a natural extension to negative numbers along the real line.

Carl Friedrich Gauss (1777-1855) Gauss had a profound influence in many fields of mathematics and science Ranked beside Euler, Newton and Archimedes as one of history's greatest mathematicians.

The fundamental theorem of algebra (1799) Every complex polynomial of degree n has exactly n roots (zeros), counted with multiplicity. (where the coefficients a0,..., an−1 can be real or complex numbers), then there exist complex numbers z1,..., zn such that If:

Complex functions

Gauss began the development of the theory of complex functions in the second decade of the 19th century He defined the integral of a complex function between two points in the complex plane as an infinite sum of the values ø(x) dx, as x moves along a curve connecting the two points Today this is known as Cauchy’s integral theorem

Augustin Louis Cauchy (1789-1857) French mathematician an early pioneer of analysis gave several important theorems in complex analysis

Cauchy integral theorem Says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same. A complex function is holomorphic if and only if it satisfies the Cauchy-Riemann equations.

The theorem is usually formulated for closed paths as follows: let U be an open subset of C which is simply connected, let f : U -> C be a holomorphic function, and let γ be a path in U whose start point is equal to its end point. Then

Georg Friedrich Bernhard Riemann (1826-1866) German mathematician who made important contributions to analysis and differential geometry

Cauchy-Riemann equations Let f(x + iy) = u + iv Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the Cauchy-Riemann equations and

The use of complex numbers today In physics: Electronic Resistance Impedance Quantum Mechanics …….

u = V =