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

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.

Area and perimeter problems

• Collection of taxes• Right angled triangle

• Perimeter = 12 units• Area = 7 square units ?

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

Cardano’s formula

3 233 23

2)2()

3(

2)2()

3(

qqpqqpx

03 qpxx

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

04153 xx

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)

(Cont.)Cardano’s formula gives:

33 21212121 xBombelli found that:

2121213

2121213

WHY????

(Cont.)

1i

812622323)2( 2332233 iiiiiii

12 i

2118126)2( 3 iiii

1211111111 2 i

2121)2( 3 i

212)2(2121 3 33 ii

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

Euler's formula in complex analysis

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

Wessel diagram / plane

Om directionens analytiske betegning

• Vector addition

Om directionens analytiske betegning

• Vector multiplicationAn example:

12

11sin

12

11cos81 iz

12

19sin

12

19cos

2

12 iz

(Cont.)The modulus is: 4

2

18

The argument is:

22

2

5

12

30

12

19

12

11

iiizz 4)0(42

sin2

cos421

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.

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.

The fundamental theorem of algebra (1799)

Every complex polynomial of degree n has exactly n roots (zeros), counted with multiplicity.

(where the coefficients a0, ..., an−1 can be real or complex numbers), then there exist complex numbers z1, ..., zn such that

If:

Complex functions

• 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)

• French mathematician

• an early pioneer of analysis

• gave several important theorems in complex analysis

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.

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)

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

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

In physics:

Electronic

Resistance

Impedance

Quantum Mechanics

…….

yxyiyxxzh 261323 22 1323 22 yxxu =

V = yxy 26