Download - Complex numbers and function - a historic journey (From Wikipedia, the free encyclopedia)
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Complex numbers and function
- a historic journey
(From Wikipedia, the free encyclopedia)
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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
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Contents (cont.)
• Complex functions
• Augustin Louis Cauchy
• Georg F. B. Riemann
• Cauchy – Riemann equation
• The use of complex numbers today
• Discussion???
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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.
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Area and perimeter problems
• Collection of taxes• Right angled triangle
• Perimeter = 12 units• Area = 7 square units ?
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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!!!!!
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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
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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
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Cardano’s formula
3 233 23
2)2()
3(
2)2()
3(
qqpqqpx
03 qpxx
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Rafael Bombelli
Made translations of Diophantus’ books
Calculated with negative numbers
Rules for addition, subtraction and multiplication of complex numbers
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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)
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(Cont.)Cardano’s formula gives:
33 21212121 xBombelli found that:
2121213
2121213
WHY????
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(Cont.)
1i
812622323)2( 2332233 iiiiiii
12 i
2118126)2( 3 iiii
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1211111111 2 i
2121)2( 3 i
212)2(2121 3 33 ii
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Rene Descartes (1596 – 1650)
• Cartesian coordinate system
• a + ib
• i is the imaginary unit
• i² = -1
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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
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Leonhard Euler 1707 - 1783
• Swiss mathematician• Collected works fills
75 volumes• Completely blind the
last 17 years of his life
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Euler's formula in complex analysis
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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
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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
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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
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Wessel diagram / plane
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Om directionens analytiske betegning
• Vector addition
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Om directionens analytiske betegning
• Vector multiplicationAn example:
12
11sin
12
11cos81 iz
12
19sin
12
19cos
2
12 iz
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(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):
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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.
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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.
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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:
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Complex functions
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• 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
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Augustin Louis Cauchy (1789-1857)
• French mathematician
• an early pioneer of analysis
• gave several important theorems in complex analysis
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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.
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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
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Georg Friedrich Bernhard Riemann (1826-1866)
• German mathematician who made important contributions to analysis and differential geometry
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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
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The use of complex numbers today
In physics:
Electronic
Resistance
Impedance
Quantum Mechanics
…….
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yxyiyxxzh 261323 22 1323 22 yxxu =
V = yxy 26