CSC - SY

We can find easily that: sum(1 / n^2) < 2 by using integration,
but can you post how can we find the exact value by using complexes

That is true in this special case, since the function 1/x^2 is a decreasing positive function on the

positive domain [1..infinity].

But in general this is a falacy.

I'd like to show how we compute the summation, the forum doesn't support mathematical expressions

but I will sketch the stepsm and the interested reader canexplore the details in the link below.

Note: The solution is not that easy... It needs a well knowledge of complex analyis....

Review: Cauchy Residue Theorem

The integral of a complex function h(z) along a countour Gamma is 2.Pi.i times the sum of the

residues of the singularities in the interior of the countour...

The idea behind the method below is to exhibit the desired summation as a part of the summation

mentioned in the theorem.

So our task is to find a complex function which has only simple poles at the integers and whose

residue is 1 there.

The fuction F(z) = cot(z) has simple poles for z = n.pi (for all integers n)...

So the desired function is pi.cot(pi.z)

We multiply this function by the function R(z) = 1/z2 to get the function g(z)

Applying the theorem on g(z) and a square contour Gamma(N)
centered at the origin and whose side length is 2N + 1 for an integer N
yields an equation whose:
LHS is a complex integeral
RHS is S1 + S2:
S1 = Sum(R(n), n: -N -> +N and n <> a pole or R(z))
S2 = Sum(Res(f, z(k)), z(k) is a pole of R inside the contour Gamma(N)).

The LHS vanishes to zero when N -> inf

So

S1 = - S2

But we want Sum(1/n2, 1, inf) which is simply S1 / 2 or -S2 /2

just one task is left: somputing S2

Taking the Laurent expansion of f(z) = pi.cot(pi.z) around 0 we get

1/z - pi^2 /3 . z - pi ^ 2 /45 . z^3 + O(z^5)

and the S2 here is the coffiecient of z that is - pi*pi / 3

There for S = Sum(1/n2, n:1 -> inf) = pi * pi /6

Conclusion:
===========

1- The solution may appear very taugh and messy, but a mathematician who master the complex analysis

and its powerful theorem find it typical.

2- There is a wide range of maneuver techniques in the complex space, while the real one has much

less maneuver techniques.

Computing the precious summaion without using the complex space is also taugh. IT deals extensively

with the theory of functions and with Euler (pronounced oiler since he is Austrian) integerations

and....

For me mathematics is great and fantastic! Although physics may come out with applications directly

related to our every day life and to the industerial revolution, but advanced physics depend heavily

on math.

Complex analysis is very very powerful and active in mathematics.
Fourier analysis (series and trandform), Laplace and Z transform depends essentialy on the complex

analysis.

I;m sorry for such along post. The mathematics team tends to discuss much attractive ideas

concerning the Applied Number Theory.

The link:
======

http://www.maths.ed.ac.uk/~jmf/English/MT3/ComplexAnalysis.pdf

Regards[/url]

A great post Bilal, thank you very much, you really saved me the time looking in the residue cauchi's theorem...brilliant.

Anyway, for a given series: finding an upper bound is very easy and can be applicated with any series IF it converges to some value, as I said before by integration, but finding the exact function requires some mathematical weapons.....

Sorry Bilal, but you posted a link to a page that can't be found, please provide us with other e-books about complex space

Thank you firstly for the subject, i can't open the file too, so i want to give me another e_book related with it
thank you twice

كتب "derarief":

Thank you firstly for the subject, i can't open the file too, so i want to give me another e_book related with it
thank you twice

Hey derarief

I just investigated the matter, and seems that the site was rearranged.
So here's the new link to the document:
http://www.maths.ed.ac.uk/~jmf/Teaching/MT3/ComplexAnalysis.pdf

Also check out the page: http://www.maths.ed.ac.uk/~jmf/Teaching/MT3.html

Some interesting documents in there.

i download it strontium90, thank you