Integration

Calculus (Integrals, Series)
Post Reply
Papapetros Vaggelis
Community Team
Posts: 426
Joined: Mon Nov 09, 2015 1:52 pm

Integration

#1

Post by Papapetros Vaggelis »

Calculate, if it exists, the integral

\(\displaystyle{\int_{0}^{\infty}\dfrac{x^3}{e^x-1}\,\mathrm{d}x}\)
mathofusva
Posts: 33
Joined: Tue May 10, 2016 3:56 pm

Re: Integration

#2

Post by mathofusva »

Let the proposed integral be $I$. We show that $I = \pi^4/15$. Indeed,
\begin{eqnarray*}
I & = & \int_0^\infty\,\frac{x^3e^{-x}}{1 - e^{-x}}\,dx\\
& = & \int_0^\infty\,\sum_{n=0}^\infty\,x^3e^{- (n+1)x}\,dx\\
& = & \sum_{n=0}^\infty\,\int_0^\infty\,x^3e^{- (n+1)x}\,dx\\
& = & \sum_{n=0}^\infty\,\frac{1}{(n+1)^4}\,\int_0^\infty\,t^3e^{-t}\,dt\\
& = & 6\,\zeta(4) = \frac{\pi^4}{15}.
\end{eqnarray*}
Here the exchange order of the integral and summation is justified by the monotone convergence theorem, since the summands are all positive.
User avatar
Riemann
Posts: 176
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

Re: Integration

#3

Post by Riemann »

Of course we are dealing with a Mellin Transform here. In general it holds that:

$$\int_{0}^{\infty} \frac{x^{s-1}}{e^x-1} \, {\rm d}x = \Gamma(s) \zeta(s) , \; s \in \mathbb{C} \mid s>1$$

And for future reference

$$\int_{0}^{\infty} \frac{x^{s-1}}{e^x+1} \, {\rm d}x = \left\{\begin{matrix}
\Gamma(s) \eta(s) &, & s \in \mathbb{C} \mid s>1 \\
\ln 2& , & s=1
\end{matrix}\right.$$
Hidden Message
The first formula is proved in an exact way as mathofusva did. The second one is proved similarly. Here $\eta$ denotes eta Dirichlet's function.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
Post Reply

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 9 guests