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Infinite Product in partition function 
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#1
Mar2514, 03:52 PM

P: 205

Hi I am having trouble following the derivation of the fermionic oscillator partition function in zeta function regularisation. Specifically the following step:
[itex] Z( \beta ) = e^{ \beta \omega /2 } lim_{ N \rightarrow \infty } \prod_{ k = N/4 }^ {N/4 } \left[ i( 1\epsilon \omega ) \frac{ \pi (2k1) }{ \beta } + \omega \right] [/itex] [itex] = e^{ \beta \omega /2 }e^{  \beta \omega /2} \prod_{k = 1}^{ \infty } \left[ ( \frac{ 2 \pi (k1/2)}{\beta} )^{2} + \omega^{2} \right] [/itex] where [itex] \epsilon = \frac{\beta}{N} [/itex] 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 


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