## 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 … Läs mer

Publicerat i matematik 1c, Philosophy of Science | | 1 kommentar

## Chinas first aircraft-carrrier

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## 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 … Läs mer

## 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 … Läs mer

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## 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 … Läs mer

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## 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 … Läs mer

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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 … Läs mer

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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) … Läs mer

## 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 … Läs mer

Publicerat i Imaginary numbers, matematik 4, matematik 5 | | Lämna en kommentar

## 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 … Läs mer

Publicerat i matematik 1c, matematik 4, matematik 5 | | 2 kommentarer