# Infinite Product in partition function

by sgd37
Tags: function, infinite, partition, product
 P: 205 Hi I am having trouble following the derivation of the fermionic oscillator partition function in zeta function regularisation. Specifically the following step: $Z( \beta ) = e^{ \beta \omega /2 } lim_{ N \rightarrow \infty } \prod_{ k = -N/4 }^ {N/4 } \left[ i( 1-\epsilon \omega ) \frac{ \pi (2k-1) }{ \beta } + \omega \right]$ $= e^{ \beta \omega /2 }e^{ - \beta \omega /2} \prod_{k = 1}^{ \infty } \left[ ( \frac{ 2 \pi (k-1/2)}{\beta} )^{2} + \omega^{2} \right]$ where $\epsilon = \frac{\beta}{N}$ I understand that we multiply the negative and positive parts of the product and take the limit of N however k going to -k gives rise to cross terms that I don't know how to deal with. Thanks for your help