Closed-Form Solution for f(A,B) Equation

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    Closed-form solution
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Discussion Overview

The discussion revolves around the search for a closed-form solution for the equation f(A,B) = ∑_{n∈ℤ} e^{-Bn² + iAn}, where A is a real number and B is positive. The context includes mathematical reasoning and exploration of related concepts.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant inquires about the existence of a closed-form solution for the given equation.
  • Another participant suggests that this problem is a more challenging version of a previously discussed question, indicating that an answer was not found in that earlier discussion.
  • A different participant expresses disbelief at the lack of a solution, sharing their experience with calculating the density operator for a particle on a circle, which involves similar exponential terms.
  • Another participant asserts that the problem is indeed exotic and suggests the use of elliptic theta functions as a potential approach.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the nature of the problem, with some expressing skepticism about the difficulty of finding a solution and others emphasizing its complexity.

Contextual Notes

There are references to related mathematical concepts, such as the density operator and elliptic theta functions, but the discussion does not resolve the mathematical steps or assumptions involved in the original equation.

tom.stoer
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Does anybody know a closed for

[tex]f(A,B) = \sum_{n\in\mathbb{Z}}e^{-Bn^2 + iAn}[/tex]

with A real and B positive.
 
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I can't believe that.
I tried to calculate the density operator in position rep. for a particle on a circle. You get the exp(iAn) term from the eigenfunctions on the circle; A contains the angle. And you get the exp(-Bn2) from exp(-H/T).
So it's nothing exotic.
 

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