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zoorna Feb27-12 08:36 AM

a simple problem in contour integration
 
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 .

zoorna Feb27-12 11:07 AM

Re: a simple problem in contour integration
 
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 .

zoorna Feb27-12 12:46 PM

Re: a simple problem in contour integration
 
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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