Zeta function for Debye Low Temp. Limit

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SUMMARY

The discussion focuses on the evaluation of the integral related to the Debye low temperature limit for specific heat, specifically ∫(x^4 e^x)/(e^x-1)^2 dx from 0 to ∞, which results in 4π^2/15. The user initially struggled to relate this integral to zeta functions due to the presence of e^x in the numerator. However, they later identified that integrating the function ∫x^3/(e^x-1) is more straightforward and can be solved using integration by parts, yielding the result π^2/15.

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  • Understanding of integral calculus, specifically integration by parts.
  • Familiarity with the Debye model for specific heat in solid-state physics.
  • Knowledge of zeta functions and their applications in mathematical physics.
  • Basic concepts of thermodynamics related to specific heat capacities.
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  • Study the properties and applications of the Riemann zeta function in physics.
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Physicists, mathematicians, and students studying thermodynamics and statistical mechanics, particularly those interested in the Debye model and integral evaluations in low-temperature physics.

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Hello,

In the low temp. limit of Debye law for specific heat, we encounter the following integral:

∫(x4 ex)/(ex-1)2 dx, from 0 to ∞.

The result is 4∏2/15.

I have searched and found this to be related to zeta function but zeta functions do not have ex in the numerator so I am unable to evaluate the integral using them.

Any help?
 
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Never mind. I found that what should be integrated is the energy, ∫x3/(ex-1) which can be integrated easily using integration by parts then using zeta function, the result is π2/15.
 

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