Sum of exponential of (-a^2 n^2)

Therefore, the actual result is a combination of these two asymptotic behaviors, which can be expressed as f(a)=sqrt(pi)/2a+g(a) where g(a) is a function that does not contain any terms of the form a^(-m) or a^m, ensuring that the result is always real and does not become complex. This problem arises in the calculation of the partition function of a particle in a 1-D box, and the final result is sqrt(pi)/2a. In summary, the conversation discusses the calculation of the sum of exponential of a quadratic function, which can be analytically expressed as a hypergeometric function. The result of the sum is a combination of two asymptotic behaviors,
  • #1
sanjibghosh
50
0
I am in a trouble to work out the sum, i.e.
"sum of exponential of[-a^2.n^2]" , where, n is an integer... and sum over 1 to infinity, and "a" is "not small...", I've calculate two asymptotic behaviours , i,e when "a<<1" , in this case I've integrated it and have got, sqrt(pi)/2a-1, and in the limit a->infinity, it is simply "0", so the actaul result may be,
f(a)=sqrt(pi)/2a+g(a), but if g(a) contains a^(-m), where m>1, then actually this would have dominated in the limit a->0, but its not the case... so g(a) cannot contain the term...a^(-m), now if g(a) contain a term like "a^m", m>1, then in the limit a->infinity we'd have got "infinity" instead of "0", so this is also not possible... and if here in "a^(-m) "m" is such that 1>m>0, then we will get a complex result...for -ve "a" but that is forbidden... as each term is real... how can the sum become complex... so I guess the result should be..."sqrt(pi)/2a"... please help... (this problem occurred when I was calculating the partition function of particle in an 1-D box...)
 
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  • #2
The result should indeed be sqrt(pi)/2a. The issue here is that you are summing the exponential of a quadratic function, which can be analytically expressed as a hypergeometric function. This means that the result of the sum is not simply a polynomial in a, but a more complicated expression. However, for large values of a, the sum converges to 0, and for small values of a it converges to sqrt(pi)/2a.
 

Related to Sum of exponential of (-a^2 n^2)

1. What is the purpose of the sum of exponential of (-a^2 n^2)?

The sum of exponential of (-a^2 n^2) is a mathematical function used to calculate the total value of a series of exponential terms with a common base (-a^2), where the exponent (n) increases by a constant value. It can be used in various fields such as physics, engineering, and finance to model and analyze processes that involve exponential growth or decay.

2. How is the sum of exponential of (-a^2 n^2) calculated?

The sum of exponential of (-a^2 n^2) can be calculated using the formula: ∑ e^(-a^2 n^2) = (1/2a) * √π * [1 + erf(a)], where erf is the error function. This formula can be derived using a variety of mathematical techniques, such as integration and series expansion.

3. What are the applications of the sum of exponential of (-a^2 n^2)?

The sum of exponential of (-a^2 n^2) has various applications in different fields. In physics, it is used to model the behavior of particles in a potential well. In engineering, it is used to study the response of systems to a series of impulses. In finance, it is used to model the value of financial assets over time.

4. Can the sum of exponential of (-a^2 n^2) be approximated?

Yes, the sum of exponential of (-a^2 n^2) can be approximated using various methods, such as numerical integration, Taylor series, and asymptotic expansions. The accuracy of the approximation depends on the chosen method and the value of the parameters.

5. Are there any real-world examples of the sum of exponential of (-a^2 n^2)?

Yes, there are many real-world examples of the sum of exponential of (-a^2 n^2). One common example is the study of the spread of diseases, where the number of infected individuals follows an exponential growth or decay pattern. Another example is the study of radioactive decay, where the number of decaying atoms also follows an exponential pattern.

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