1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Deriving the Stefan-Boltzman law and integration tricks

Tags:
  1. Aug 6, 2015 #1
    1. The problem statement, all variables and given/known data
    HI people,

    I was trying to derive the stefan-boltzman law from the planc's formula, I kind of got stuck with an integral



    2. Relevant equations

    $$ \int_{0}^{\infty} \frac{x^3}{e^x -1} dx $$
    3. The attempt at a solution
    I tried simplifying it with

    $$ \int_{0}^{\infty} x^3 e^{-x} \sum_{n=0}^{\infty} e^{-nx} dx $$

    Now I don't know what to do with the summation. I could evaluate

    $$ \int_{0}^{\infty} x^3 e^{-x} dx = 6$$

    pleas help me to get the rest of it, I see the answer comes with $\pi$, how do I get it in this kind of equation? Is there any special trick to solve these type of integrals?
    thanks in advance.
     
  2. jcsd
  3. Aug 6, 2015 #2

    ShayanJ

    User Avatar
    Gold Member

    Under the condition that both the series and the integral converge, you can swap them. Here we're sure that the series converges because we got it by taylor-expanding a function. About the integral, we have a polynomial against an exponential with negative exponent which means converges too. So we have ##\displaystyle \sum_{n=0}^\infty \int_0^\infty x^3 e^{-x}e^{-nx} dx=\sum_{n=0}^\infty \int_0^\infty x^3 e^{-(n+1)x}##. Now you if you integrate by parts three times(each time considering u to be the algebraic factor and dv to be the exponential one), you can obtain the solution to the integral.
     
  4. Aug 7, 2015 #3
    Ah great, thank you very much, I almost got it on the track now, this is

    $$ \int_{0}^{\infty} x^3 e^{-(n+1)x} dx = 6 (n+1)^{-4}$$

    can you just tell me a little about how to evaluate the summation of $$\sum \frac{1}{(n+1)^4}$$ though, I think I forgot this evaluation.

    thanks in advance
     
  5. Aug 7, 2015 #4

    ShayanJ

    User Avatar
    Gold Member

    The series ## \displaystyle \sum_{n=0}^\infty \frac{1}{(n+1)^4}## can be written as ## \displaystyle \sum_{n=1}^\infty \frac{1}{n^4}## which is equal to ## \zeta(4) ##, where ## \zeta(s) ## is the Riemann zeta function. So you don't need to evaluate the series, just look for tables of the values for this function or compute it using math softwares.
     
  6. Aug 8, 2015 #5
    problem solved, thanks a lot
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Deriving the Stefan-Boltzman law and integration tricks
  1. Power law trick DE (Replies: 3)

  2. Integration Tricks (Replies: 13)

  3. Trick Triple Integral (Replies: 7)

Loading...