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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Complex numbers and function

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Complex numbers and function

- a historic journey

(From Wikipedia, the free encyclopedia)


  • 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.

Collection of taxes

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

Cardano’s formula

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)


Cardano’s formula gives:

Bombelli found that:



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

Vector addition

Om directionens analytiske betegning

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.

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)

Thefundamental 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

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


The use of complex numbers today

In physics:




Quantum Mechanics


u =

V =

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