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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 of drawing logical conclusions from already proven results.  This chain of reasoning can then be traced all the way back to the fundamental axioms.

Kant pointed out that mathematics is a purely theoretical science.

The axiomatic system of  mathematics is independent of the physical world. Still mathematics is of outmost importance for the empirical sciences since it provides a means for constructing mathematical models able to accurately describe and predict outcomes in the  real world.

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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 it is the final stage of factorization of numbers.

Also polynomials can be factorized with e.g. the conjugate rule

x4 -1 = (x2-1)(x2+1) = (x-1)(x+1)(x2+1)

or with the theorem of factorization stating that the roots of a polynomial are also factors of that polynomial.

Ex.

Consider the polynomial p(x) = (x-2)(x-3).

As can easily be seen the roots of p(x) are x = 2 and x = 3.

Multiplying the parentheses results in

p(x) = x2 – 5x + 6.

Therefore

x2 – 5x + 6 = (x-2)(x-3)

Therefore the roots to this polynomial are also the factors of it!

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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 Pascal’s triangle. This triangle isbased on the principle that an element in it is the sum of the nearest elements above it.

The elements in Pascal’s triangle turns out to be the binomial-coefficients. This means that they are identical to the numbers you get when you expand binomials with different exponentials. A binomial is the sum of two numbers (x + y) .

As an example we can take (x+y)^2 = x^2 + 2xy + y^2.

The coefficients here can be found in the second row of the periodic system: 1    2    1.

The  coefficients for (x + y) ^3 is given by the third row: 1  3  3   1 and so on.

This is spectacular in its own right but it doesn’t   end there: The binomialcoefficients also have a combinatorical interpretation: They give the  number of combinations if you e.g. are to select k objects out of n possible.

Example: Suppose you wish to pick 2  apples out of a set of three. You find the answer as the second element in the third row: 3.

This is called the number of combinations possible when selecting two elements from a set of three. In a combination the order is irrelevant.   According to convention it is writen as

Here n! = the number of permutations of n i.e. the number of possible arrangements of n objects when the order is important, k! is the number of permutations of k objects and (n-k)! the number of permutations of the difference of n and k.
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Theorem

Suppose U is an open subset of the complex plane C, f : UC is a holomorphic function and the closed disk D = { z : | zz0| ≤ r} is completely contained in U. Let be the circle forming the boundary of D. Then for every a in the interior of D:

where the contour integral is taken counter-clockwise.

The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. Since the reciprocal of the denominator of the integrand in Cauchy’s integral formula can be expanded as a power series in the variable (a − z0), it follows that holomorphic functions are analytic. In particular f is actually infinitely differentiable, with

This formula is sometimes referred to as Cauchy’s differentiation formula.

The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure.

This proof is based on the fact that the value of the integral is invariant under deformation as long as the integrand is analytic over this closed area.

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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 are the same.

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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 Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a)

and

(1b)
Proof:
Suppose that

is a function of a complex number z. Then the complex derivative of ƒ at a point z0 is defined by

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

On the other hand, approaching along the imaginary axis,

The equality of the derivative of ƒ taken along the two axes is

Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of (this is called a domain in ).
function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.
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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) =z^2 and z = x + iy gives x^2 + 2ixy + y^2 with real part x^2 + y*2 and imaginary part 2xy.

A necessary condition for a function of a complex variable to be differentaible is that it satisfies the Cauchy -Riemann equations.

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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 called the Cartesian representation. (Rene Descartes)

From the figure below it is evident that

x = r cosφ

y = r sin φ  and thus z = r( cosφ + i sinφ) . This is called polar representation of the complex number z. r is the modulus of the vector z (i.e. its length) and φ is called the argument of z.    The modulus can be  computed by multiplying the  complex number z with its conjugated complex number z

By adding the  Taylor series for cos(x) and  i sin(x) we get the series expansion for the exponential function eix. This relation is called Euler’s formula.

This leads to the exponential representation of a complex number:

z= r e(iφ)

This

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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 to take the square root of negative numbers.

They are represented along an axis horizontal to the axis of real numbers.

A combination of a real number and a imaginary number is called a complex number.

z = 2  + 3i.