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

1. Feb 14, 2009

### sanjibghosh

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".... plz help..... (this problem occured when I was calculating the partition function of particle in an 1-D box....)