Simplifying a double summation

In summary, the conversation discusses simplifying a function involving sums and error functions. The possibility of reducing the function to a simpler form is mentioned, along with the suggestion of using integrals and examining the arguments of the error functions for potential simplification.
  • #1
Ad VanderVen
169
13
TL;DR Summary
Simplifying a double summation.
Is it possible to simplify the function below so that the sums disappear.
$$\displaystyle g \left(x \right) \, = \, \sum _{j=-\infty}^{\infty} \left(-A +B \right) \sum _{k=-\infty}^{\infty} \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -k \right)^{2}}{\sigma ^{2}}}~\left(U -V \right)}{\sigma ~\sqrt{\pi }}$$
with
$$\displaystyle A\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j-1/2+{\it omicron} \right) }{\rho}}\right),$$
$$\displaystyle B\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j+1/2+{\it omicron} \right) }{\rho}}\right),$$
$$\displaystyle U\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k+1 \right) }{\tau}}\right)$$
and
$$\displaystyle V\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k-1 \right) }{\tau}}\right)$$
 
Last edited:
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  • #2
It seems very complex to me. Why do you estimate or expect that it would be reduced to a no sum form ?
 
  • #3
anuttarasammyak said:
It seems very complex to me. Why do you estimate or expect that it would be reduced to a no sum form ?
I do not know.
 
  • #4
Get rid of all the constants that can be taken out of the sums, they just blow up the expression for absolutely no reason.

You can express all the error functions as integrals and then see if adjacent terms have some nice relation for the boundaries that allows summation. The error function arguments look like there might be something you can combine.
 

What is a double summation?

A double summation is a mathematical concept that involves adding up a series of numbers or terms that are arranged in a two-dimensional grid or matrix. It is denoted by the symbol ∑, and is often used in statistics, physics, and other fields of science.

How do you simplify a double summation?

To simplify a double summation, you can use various mathematical techniques such as rearranging terms, using known formulas or identities, and applying properties of summations. The goal is to reduce the expression to a simpler form that is easier to evaluate or understand.

What are the benefits of simplifying a double summation?

Simplifying a double summation can make complex mathematical expressions easier to understand and work with. It can also help to identify patterns or relationships between terms, and can make it easier to calculate the final result.

Can a double summation be simplified to a single summation?

Yes, it is possible to simplify a double summation to a single summation. This can be done by using properties of summations, such as the distributive property, to combine the two summations into one. However, this is not always possible and depends on the specific terms and expressions involved.

Are there any common mistakes to avoid when simplifying a double summation?

One common mistake when simplifying a double summation is forgetting to apply the correct properties or formulas. It is important to carefully check each step and make sure that the simplification is valid. Another mistake is not properly accounting for the limits of the summation, which can lead to incorrect results.

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