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Complex numbers and functionPowerPoint Presentation

Complex numbers and function

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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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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.

Right angled triangle

Perimeter = 12 units

Area = 7 square units

Area and perimeter problems?

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

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)

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

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

Om directionens analytiske betegning

Om directionens analytiske betegning

- Vector multiplication
An example:

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.

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 and science 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

Complex functions and science

- Gauss began the development of the and science 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) and science

- French mathematician
- an early pioneer of analysis
- gave several important theorems in complex analysis

Cauchy integral theorem and science

- 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 endpoint. Then

Georg Friedrich Bernhard Riemann 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(1826-1866)

- German mathematician who made important contributions to analysis and differential geometry

Cauchy-Riemann equations 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

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 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

In physics:

Electronic

Resistance

Impedance

Quantum Mechanics

…….

u = 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

V =

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