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A simple problem in contour integration

  1. Feb 27, 2012 #1
    greetings . this is my first post here . i am preparing myself for a complex analysis course that i will be taking next semester . i came across this problem , which is probably a very simple one , but i don't know how to go about it , so bare with me:biggrin:

    we have the contour integration
    [tex] I(x)=\frac{1}{2\pi i} \int_{\sigma-iT}^{\sigma+iT}\left(\frac{x}{n}\right)^{s}ds[/tex]

    where [itex] \Re(s) ,\sigma > 1[/itex]

    x is a variable , and n is a constant .

    i need the integration to yield a constant if [itex] x=n [/itex], and zero otherwise !?!?

    my guess is that the function [itex] \frac{x}{n} [/itex] should be somehow modified to yield the desired result - constant for x=n , zero other wise - . or is it the contour that should be changed !?!?

    your help is appreciated .
     
    Last edited: Feb 27, 2012
  2. jcsd
  3. Feb 27, 2012 #2
    i'll try to make the question a bit clearer . we have the dirichlet series :

    [tex]I(s)= \sum_{n=1}^{\infty}\frac{\alpha(n)}{n^{s}} , \Re(s)>1[/tex]

    where [itex]\alpha(n) [/itex] is some arithmetic function of n .

    now i am trying to use mellin transform, or any kind of transform akin to that of fourier's, to extract [itex] \alpha(n)[/itex] . i was hoping for a kernel - function of s - that is orthogonal to all terms except for the one containing the integer i want to extract [itex] \alpha(n)[/itex] for . meaning , i am trying to find a function [itex] f(x,s) [/itex] such that :

    [tex]
    < f(x,s),n^{-s}>=\left\{\begin{matrix}
    k & , & x=n & \\
    0 & , & o.w &
    \end{matrix}\right.
    [/tex]

    k is a constant

    i hope this makes it clearer .
     
    Last edited: Feb 27, 2012
  4. Feb 27, 2012 #3
    i think i found it . using Perron's formula :

    [tex] \alpha(n)=\frac{1}{2\pi i}\int_{2-i\infty}^{2+i\infty}I(s)\frac{\left(n+1/2\right)^{s}-\left(n-1/2 \right )^{s}}{s}ds[/tex]
     
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