/sci/ - Science & Math

(cont.)

Also, the way of taking the sum to the complex plane looks somewhat like this:

First he observes that a integration variable substitution x -> n·x in the definition of the Gamma function

[math] \Gamma(s) := \int_0^\infty x^{s-1} e^{-x}\,dx [/math]

let's you write

[math] n^{-s} = \frac {1} {\Gamma(s)} \int_0^\infty x^{s-1} e^{-nx}\,dx [/math]

and then, recognizing the geometric series

[math] \sum_{n=1}^\infty (e^{-x})^n = \frac{1} { e^{-x}-1} = \frac{1} {x} - \frac{1}{2} + \frac{1}{12}x+O(x^2) [/math]

([math] \frac{1}{12} [/math], see >>7746983)

give you

[math] \sum_{n=1}^\infty n^{-s} = \frac{1} { \Gamma(s) } \int_0^\infty \frac{x^{s-1}} { e^x-1} \, {\mathrm d}x [/math]

He takes the integral into the complex plane, where he [math] \frac{1} { e^x-1} [/math] diverges periodically in steps of [math] 2\pi\,i [/math].
Playing around with the exponential function and doing some mirroring when you have symmetris, he discovers that the function obeys a reflection formula

[math] \zeta(s) = (2\, \pi)^s\,(\sin (\frac {\pi s} {2} )/\pi)\,\Gamma(1-s)\ \zeta(1-s) [/math]

and now for negative s you know the value once you know it for positive ones. E.g.

[math] \zeta(-1) = (2\, \pi)^{-1}\,(\sin (\frac {\pi (-1)} {2} )/\pi)\,\Gamma(1-(-1)) \,\zeta(1-(-1)) [/math]

[math] = \frac {1} {2} \frac {1} {\pi^2} \, (-1) \, 1! \, \sum_{n=1}^\infty \frac {1} {n^2} [/math]

[math] = -\frac {1} {2} \frac {1} {\pi^2} \frac {\pi^2} {6} [/math]

[math] = -\frac {1} {12} [/math]

PS: I go to bed now, but let me just point at a comprehensive (100pages??) "study guide" that shortly discusses the possible standard logic books on all level, and gives a guide where to start:

http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic2015.pdf

I personally like those notes for logical rules and proof (pictures above are from there)
http://www.personal.psu.edu/t20/notes/
(first link)

What level of sophistication are you interested in?
I remember writing down an analogy here

http://archive.foolz.us/sci/thread/4688051/#q4688164

The explanation suggests how the proof should look like - the tricky part is to implement the self-reference before you do anything.

Formally, a proof is a sequence of sentences and after you map the sentences to numbers (Gödel numbering), you can reason about proofs by reasoning about numbers.
The subject of arithmetic comes into play when Gödel shows how to use it to encode not just sentences but sequences of sentences as numbers (the β function) (and one could argue he introduces programming along the way, primitive recursion is a technical keyword). The incompleteness theorem then comes from a no-go (diagonal) argument invoving those numbers.

http://en.wikipedia.org/wiki/G%C3%B6del_numbering
http://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function
http://en.wikipedia.org/wiki/Diagonal_lemma

Here's a book on the subject I like: www.logicmatters.net/igt/

>>6644151
>>6644200
not gonna defend any notation here, but for clarification:
It's a multiple integral (not iterated), e.g.
<span class="math">\int\int d^3k[/spoiler] is <span class="math">\int dk_1\int dk_2\int dk_3[/spoiler], where <span class="math">k_i[/spoiler] is the i'th variable.
As the other guy said, <span class="math">x^2[/spoiler] is the inner product of x with itself
and finally and one writes d and not n because d is, roughly speaking, a complex number

http://en.wikipedia.org/wiki/Dimensional_regularization

@OP: I forgot how to approach these, but if it's really important I can look it up tomorrow (dig out notes - don't make me do it if it's not really relevant plz.)

For people crossing by Vienna, I saw there's an exhibition starting:

Ausstellung
„Kurt Gödel und die Ursprünge der Logik in Wien“
Akademie der Bildenden Künste Wien.
Täglich 15. Juli – 24. Juli; 10.00-18.00;
Eintritt frei

seen here
http://derstandard.at/2000002837430/Ausstellung-ueber-Kurt-Goedel-Die-Logik-der-Unvollstaendigkeit