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We begin the solution of this integral by replacing the denominator with its infinite series equivalent:

(Eq'n 1)

So we have as our integral

(Eq'n 2)

To clarify that integration we make the substitution y=mx, so we have

(Eq'n 3)

The infinite series multiplying our simplified integral is the Dirichlet series expansion of the Riemann zeta function,

(Eq'n 4)

so we have

(Eq'n 5)

Now we can integrate by parts. We have

(Eq'n 6)

We repeat that procedure n-1 more times and get

(Eq'n 7)

Example Application:

The Stefan-Boltzmann Law

If we have a hollow body at a uniform absolute temperature T, then we know that the radiation inside that body precisely mimics the radiation emanating from a perfectly black body and that its energy density per unit of frequency (its spectral density) conforms, via Planck's theorem, to the mathematical statement that

(Eq'n 8)

To calculate the total energy density of the radiation we multiply that equation by the minuscule element of frequency and integrate it over all frequencies:

(Eq'n 9)

in which I have made the substitution q=h/kT. We thus have

(Eq'n 10)

We know the factorial of three (3!=6) and the Riemann zeta function for index four (ζ(4)=π4/90), so we have the energy density of the radiation inside our hollow body as

(Eq'n 11)


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