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# Complex numbers and function - PowerPoint PPT Presentation

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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Presentation Transcript

- a historic journey

• 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???

• 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

?

• 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!!!!!

• 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

• 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

Calculated with negative numbers

Rules for addition, subtraction and multiplication of complex numbers

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)

Cardano’s formula gives:

Bombelli found that:

WHY????

• Cartesian coordinate system

• a + ib

• i is the imaginary unit

• i² = -1

• (cosx + i sinx)^n = cos(nx) + i sin(nx)

• z^n= 1

• Newton knew this formula in 1676

• Poor – earned money playing chess

• Swiss mathematician

• Collected works fills 75 volumes

• Completely blind the last 17 years of his life

• 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

• 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

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

• Vector multiplication

An example:

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.

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 =