(edited by Roberto Bigoni)

The integrals of form

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

The introduced sum indeed is a geometric series of common ratio *e ^{-x}*.
If

Therefore

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

To calculate the integral
we can let *t=ix*; so

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

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

n | ζ(n) |
---|---|

2 | π^{2}‾‾‾ 6 |

4 | π^{4}‾‾‾ 90 |

6 | π^{6}‾‾‾‾‾ 945 |

8 | π^{8}‾‾‾‾‾‾ 9450 |

Then the integrals *I _{n}* with odd

n | I_{n} |
---|---|

1 | π^{2}‾‾‾ 6 |

3 | π^{4}‾‾‾ 15 |

5 | 8π^{6}‾‾‾‾‾ 63 |

7 | 8π^{8}‾‾‾‾‾‾ 15 |

Last Revised: May 2018