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

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 ( )
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. Cardanos 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 Cardanos 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.)
Cardanos 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 + isinx)n = cos(nx) + isin(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. Om directionens analytiske betegning
Vector addition

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 – proffesional 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. 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.

28. The fundamental 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 end point. 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 =