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Infinite Product in partition function

by sgd37
Tags: function, infinite, partition, product
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sgd37
#1
Mar25-14, 03:52 PM
P: 206
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 (2k-1) }{ \beta } + \omega \right] [/itex]

[itex] = e^{ \beta \omega /2 }e^{ - \beta \omega /2} \prod_{k = 1}^{ \infty } \left[ ( \frac{ 2 \pi (k-1/2)}{\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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