Example 9.9: BoseEinstein integrals and Riemann zeta functions (x) Show that 0 x2ex2 1

Example 9.9: Bose–Einstein integrals and Riemann zeta functions ζ(x)

Show that¶¶



∞

0 x2e−x2 1 − e−x2 dx =

1 4

√πζ



3 2



, ζ



3 2



= 2.612. (9.43)

More generally 

∞

0 xn ex − 1 dx = n! ζ(n + 1). (9.44)

Thus Bose–Einstein integrals can be related to Riemann zeta functions.

[Hint: Use the change of variables in



∞

0 x2dxe−x2

[1 + exp(−x2) + · · ·] =



∞

0 x2dx exp(−x2)[1 + 2−3/2 + · · · ]].