Complex Analysis finding a sum

In summary, Complex Analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It can be used to find sums by using techniques such as contour integration, infinite series, and residues. A contour in Complex Analysis is a curve or path in the complex plane along which a function is integrated, and an infinite series is a sum of infinitely many terms. The residue theorem is also used in Complex Analysis to find sums by stating that the sum of the residues of a function over a closed contour is equal to the value of the integral of that function over the contour.
  • #1
yxgao
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Anyone know how to find a sum of a function using the poisson summation formula and the Fourier transform. Thanks!
--yxgao
 
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  • #2
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...
 
  • #3


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!
 

1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the use of calculus, algebra, and geometry to analyze and understand the properties of complex numbers.

2. How can Complex Analysis be used to find a sum?

In Complex Analysis, a sum can be found by using techniques such as contour integration, infinite series, and residues. These methods involve manipulating complex functions and integrals to obtain the desired sum.

3. What is a contour in Complex Analysis?

A contour in Complex Analysis is a curve or path in the complex plane along which a function is integrated. It can be any closed or open curve that connects two points in the complex plane.

4. What is an infinite series in Complex Analysis?

An infinite series in Complex Analysis is a sum of infinitely many terms, where each term is a complex number. These series can be convergent or divergent, and they are used to approximate complex functions and values.

5. How is the residue theorem used in Complex Analysis to find sums?

The residue theorem states that the sum of the residues of a function over a closed contour in the complex plane is equal to the value of the integral of that function over the contour. This theorem is used in Complex Analysis to evaluate integrals and find sums of complex functions.

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