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There exist two ways to solve this integral, both rather trivial. We have the direct path,

(Eq’n 1)

To confirm that result, we start by converting our integrand into an infinite series,

(Eq’n 2)

Integrating that series, term by term, gives us our second path to the solution we want,

(Eq’n 3)

To convert that expression to the original argument we simply do this:

(Eq’n 4)

Example Application:

Free Energy in a Boson Gas

In a Bose-Einstein gas, such as helium, which does not obey the Pauli Exclusion Principle, the average number of particles that occupy a state with energy ås conforms to the statement

(Eq’n 5)

subject to the condition that

(Eq’n 6)

the total number of particles in the gas. We now wish to calculate the Helmholtz free energy of the gas, the energy that changes when we vary the temperature and/or the volume of the gas.

We have the Helmholtz free energy as

(Eq’n 7)

in which Z represents the partition function associated with the gas. Changing the energy in each state without changing the number of particles in each state gives us one way to change the Helmholtz free energy. In that case we have the differential change as

(Eq’n 8)

Making the appropriate substitution from Equation 5 and applying Equation 1 then gives us

(Eq’n 9)

Thus, if we can specify the energy in each state available to the particles in the gas, we can calculate the Helmholtz free energy.

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