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Månadsarkiv: september 2012
What is mathematics?
Perhaps after 53 lectures it’s about time that I define what I mean with mathematics? Mathematics can be defined as the science dealing with quantities, numbers and geometrical objects in particular. It is characterised by its logical method which consists … Fortsätt läsa
Publicerat i matematik 1c, Philosophy of Science
Märkt geometrical objects, Immanuel Kant, logical conclusions, Mathematics, Philosophy
1 kommentar
factorization (faktorisering)
Factorization means to decompose a number or polynomial into a product of other objects, called factors, which when multiplied together gives the original number or polynomial. Ex. The number 16 = 2*2*2*2 when factorized into prime numbers. Since prime-numbers can’t be factorized … Fortsätt läsa
Pascal’s triangle
Numerology is an old phenomenon in many cultures. It is represented both in art an buildings. Buildings are constructed according tothe Golden Section and magical quadrats can be found in eg Albrect Durer’s painting ”Melancholy”. A substantial amount of mathematics is hidden in … Fortsätt läsa
Cauchy’s integralformula
Theorem Suppose U is an open subset of the complex plane C, f : U → C is a holomorphic function and the closed disk D = { z : | z − z0| ≤ r} is completely contained in U. Let … Fortsätt läsa
Cauchy’s integral theorem
Cauchy‘s integral theorem states that an analytic function f(z) the line integral around a closed path C is zero. ∫f(z)dz = 0 . This means that the curve integrals over 2 curves with the same endpoints for an analytic function … Fortsätt läsa
Publicerat i Imaginary numbers
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The Cauchy-Riemann equations
In order for a complex function of a single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0. This condition leads to the CAuchy –Riemann equations- The … Fortsätt läsa
Publicerat i Advanced, Calculus, Imaginary numbers
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de Moivre’s formula and complex-conjugation.
(e^ix )^n = cos(nx) + i sin(nx) is called de Moivre’s formula. The formula is named after the 17 th. century French huguenot mathematician Abraham de Moivre. Also the variable in of a function can be a complex number. f(z) … Fortsätt läsa
Alternative representations of complex numbers
As mentioned in the latest post any complex number may be represented by an arrow in the complex plane. This number is unambiguously described by two numbers: its real part x and its imaginary part y. z= x+iy. This is … Fortsätt läsa
Publicerat i Imaginary numbers, matematik 4, matematik 5
Märkt complex number, Imaginary numbers, polar coordinates, rene descartes
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Imaginary numbers
A solution to the simple second-degree equation x2 + 1 =0 can not be found along the line of real-numbers. Therefore it was necessary to invent a fictive number i such that i2=-1. i.e. the imaginary numbers making it possible … Fortsätt läsa
Publicerat i matematik 1c, matematik 4, matematik 5
Märkt imaginary number, Imaginary numbers, real numbers, square root of negative numbers
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