Z, <U>, and C for Hagedorn Spectrum

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The discussion focuses on deriving the partition function Z for the Hagedorn spectrum using the integral involving energy E and the parameters B and B0. The resulting expression for Z is complex, involving multiple terms with derivatives of E and powers of ΔB. To compute the average energy <U>, the derivative of the logarithm of Z with respect to B is used, while the specific heat C is derived from the temperature derivative of <U>. The participants express concerns about the complexity of the calculations for <U> and C and seek a more efficient method or verification of their calculations for Z. The conversation highlights the challenges in statistical mechanics when dealing with the Hagedorn spectrum.
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Homework Statement
The Hagedorn Spectrum with E>=0 is ##\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}## where ##B_{0} = 1/kT_{0}## where ##T_{0}## is the Hagedorn Temperature. For ##T<T_{0}## determine the partition function Z, the average energy <U> and the specific heat C for the system.
Relevant Equations
##\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}##
So to get the partition function I do the integral ##\int \alpha E^{3} e^{(B_{0}-B)E} dE##, which substituting in ##/Delta B = B_{0} - B## is ##Z = \frac{ \alpha E^{3} e^{\Delta B E}}{\Delta B} - \frac{3 \alpha E^{2} e^{\Delta B}}{\Delta B^{2}} + \frac{6 \alpha E e^{\Delta B E}}{\Delta B ^{3}} - \frac{6 \alpha e^{\Delta B E}}{\Delta B ^{4}}##. To get <U> you just calculate ##- \frac{\partial ln(Z)}{\partial B}## and to get C you just calculate ##k \frac{\partial <U>}{ \partial T}##. The math gets pretty rough calculating <U> and gets especially rough calculating C and I'm wondering if there is a much less tedious way to get them or if I messed up calculating Z.
 
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