Complex numbers and function

1. Complex numbers and function - a historic journey (From Wikipedia, the free encyclopedia)

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

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

4. 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.

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

6. 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!!!!!

7. 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

8. 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

9. Cardano’s formula

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

11. 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)

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

13. (Cont.)

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

16. 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

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

18. Euler's formula in complex analysis

19. 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

20. 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

21. 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

22. Wessel diagram / plane

23. Vector addition Om directionens analytiske betegning

24. Om directionens analytiske betegning • Vector multiplication An example:

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

26. 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.

27. 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. Carl Friedrich Gauss (1777-1855)

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

29. Complex functions

30. 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

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

32. 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.

33. 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 endpoint. Then

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

35. 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

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

37. u = V =