Closed-form expression of an infinite sum

In summary, the conversation discusses finding a closed form for the expression S(z) = \sum_{n=1}^\infty \exp(\frac{i}{n}) z^{-n} and mentions a method of resumming summations using contour integration. The method involves constructing a function with poles at integer values and evaluating the integral using a contour. However, this method may not always work and could be as difficult as the original summation. Further examination of the summation is needed to determine its closed form.
  • #1
Manchot
473
4
Hey all,

This isn't actually for a homework problem, but I'm trying to find a closed form for the following expression:

[tex]S(z) = \sum_{n=1}^\infty \exp(\frac{i}{n}) z^{-n}[/tex]

(Provided that it converges, of course.) Anyone have any tips, or know of any transcendental functions that might help me out? If you did, that would be great.
 
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  • #2
There is a method to resum summations using contour integration. I'm not sure how helpful it is with this particular example though.

Have you heard of this method before?
 
  • #3
No, I haven't, but I think I have a guess as to what you mean. Do you construct a function whose poles are located at the integers 1 through n, and integrate around them?
 
  • #4
Yes, you usually multiply by the function [itex]\pi\cot\pi z[/itex], which has simple poles of residue 1 at integer values along the real axis. If you pick a contour which encloses the positive real axis, then the integral will give you [itex]2\pi i\times[/itex](your summation). If you then deform your contour into one which allows you to evaluate your integral, this will give you your summation in closed form. It doesn't always work though - the integral you have to evaluate could be just as difficult as the original summation. I'll have a closer look at the summation when I get more time.
 

1. What is a closed-form expression?

A closed-form expression is a mathematical expression that can be written in a finite number of standard mathematical operations, such as addition, subtraction, multiplication, division, exponentiation, and radicals.

2. What is an infinite sum?

An infinite sum is a series of numbers that continues infinitely, with each term being added to the previous term. It is also known as an infinite series.

3. Why is it important to find a closed-form expression for an infinite sum?

A closed-form expression allows us to efficiently calculate the sum without having to add an infinite number of terms. It also provides a deeper understanding of the behavior and patterns of the infinite sum.

4. How do you find a closed-form expression for an infinite sum?

To find a closed-form expression for an infinite sum, we use various mathematical techniques such as algebraic manipulation, geometric series, and the properties of limits.

5. Can every infinite sum be expressed in closed-form?

No, not every infinite sum can be expressed in closed-form. Some infinite sums are non-elementary, meaning they cannot be expressed in terms of standard mathematical operations. However, many common infinite sums, such as geometric series and telescoping series, can be expressed in closed-form.

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