How to Find a Sum Using Poisson Summation Formula and Fourier Transform?

[yxgao]

Anyone know how to find a sum of a function using the poisson summation formula and the Fourier transform. Thanks!
--yxgao

 

[ReyChiquito]

Its easier with calculus of residues, u can see a detailed explanation in the texts of Marsden or Churchill, or any other complex variable books i guess...

 

[M98Ranger]

Hello yxgao,

Yes, I am familiar with using the Poisson summation formula and the Fourier transform to find sums of functions. The Poisson summation formula is a powerful tool in complex analysis that allows us to relate the sum of a function over integers to its Fourier transform. This can be helpful in evaluating sums that are difficult to compute directly.

To use the Poisson summation formula, we first need to express our function in terms of its Fourier transform. Then, we can use the formula to rewrite the sum in terms of the Fourier transform, making it easier to compute. The formula is given by:

∑n=−∞f(n)=∑k=−∞ˆf(k)

Where ˆf(k) is the Fourier transform of f(x). This formula allows us to exchange the sum over integers with the sum over the Fourier coefficients, which can be easier to evaluate.

To use the Fourier transform to find a sum, we first need to compute the Fourier coefficients of our function. Then, we can use the formula above to compute the sum by summing over the Fourier coefficients. The Fourier transform is a powerful tool in complex analysis that allows us to decompose a function into its frequency components. This can be helpful in evaluating sums over periodic functions.

I hope this helps answer your question. If you need further assistance, please feel free to ask. Good luck with your studies in complex analysis!