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Complex numbers and function. - a historic journey. (From Wikipedia, the free encyclopedia). Contents. Complex numbers Diophantus Italian rennaissance mathematicians Rene Descartes Abraham de Moivre Leonhard Euler Caspar Wessel Jean-Robert Argand Carl Friedrich Gauss.

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complex numbers and function
Complex numbers and function

- a historic journey

(From Wikipedia, the free encyclopedia)

contents
Contents
  • Complex numbers
  • Diophantus
  • Italian rennaissance mathematicians
  • Rene Descartes
  • Abraham de Moivre
  • Leonhard Euler
  • Caspar Wessel
  • Jean-Robert Argand
  • Carl Friedrich Gauss
contents cont
Contents (cont.)
  • Complex functions
  • Augustin Louis Cauchy
  • Georg F. B. Riemann
  • Cauchy – Riemann equation
  • The use of complex numbers today
  • Discussion???
diophantus of alexandria
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

Area and perimeter problems

?

can you find such a triangle
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
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 mathematicians8
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
rafael bombelli
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
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)

slide12

(Cont.)

Cardano’s formula gives:

Bombelli found that:

WHY????

rene descartes 1596 1650
Rene Descartes (1596 – 1650)
  • Cartesian coordinate system
  • a + ib
  • i is the imaginary unit
  • i² = -1
abraham de moivre 1667 1754
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
Leonhard Euler 1707 - 1783
  • Swiss mathematician
  • Collected works fills 75 volumes
  • Completely blind the last 17 years of his life
caspar wessel 1745 1818
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
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
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
om directionens analytiske betegning24
Om directionens analytiske betegning
  • Vector multiplication

An example:

slide25

(Cont.)

The modulus is:

The argument is:

Then (by Wessels discovery):

jean robert argand 1768 1822
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.

Carl Friedrich Gauss (1777-1855)
the fundamental theorem of algebra 1799
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

slide30
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
Augustin Louis Cauchy (1789-1857)
  • French mathematician
  • an early pioneer of analysis
  • gave several important theorems in complex analysis
cauchy integral theorem
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.
slide33

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

georg friedrich bernhard riemann 1826 1866
Georg Friedrich Bernhard Riemann(1826-1866)
  • German mathematician who made important contributions to analysis and differential geometry
cauchy riemann equations
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
The use of complex numbers today

In physics:

Electronic

Resistance

Impedance

Quantum Mechanics

…….

slide37

u =

V =

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