Bose Integrals

The integrals of form

fig.1

which must be calculated, for example, in the black body theory, may be written

fig.2

The introduced sum indeed is a geometric series of common ratio e^-x. If x>0 then 0<e^-x<1 and, when the common ratio q is <1,

Therefore

fig.4

The integral of the sum is the sum of the integrals of its terms: then

fig.5

To calculate the integral fig.6 we can let t=ix; so

fig.7

The integral fig.8 is the function Γ(n+1) (Euler's Gamma) that, for natural positive integer, coincides with the factorial n!

fig.9

Now we can write

By introducing the function ζ (Riemann's zeta) (limitedly to natural arguments)

we finally have

The values of ζ for natural even arguments were calculated by Euler. Here are the first of them

Then the integrals I_n with odd n have the following values