How to deal with infinite poles in Mellin transform for sum evaluation?

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SUMMARY

The discussion focuses on addressing the challenges posed by infinite poles in the Mellin transform, specifically when evaluating sums involving the inverse transform of functions like 1/sin(πs). The participant identifies that the infinite poles arise from the denominator sin(πs) and seeks guidance on handling these issues. The Mellin inverse transform for the range 0 < s < 1 is established as 1/(π(1+x)), with variations for different ranges of s. Participants are encouraged to provide examples and textbooks that illustrate similar scenarios.

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justin_huang
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I try to use mellin transform and mellin inverse transform to evaluate the sum of some series,
however when after I get the mellin inverse transform, It seems infinite pores cause the denominator is sin(pi*s), how can I deal with this situation? could you provide some textbook with example of the same situation?
 
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The Mellin inverse transform of 1/sin(pi*s) with 0<s<1 is 1/(pi*(1+x))
or other expressions for different ranges of s.
What exactly is the function f(s)/sin(pi*s) which Mellin inverse you are searching for ?
 

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