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Homework Help: Bernoulli Numbers, Euler-Maclauring Formula - Math Methods class

  1. Dec 6, 2008 #1
    1. The problem statement, all variables and given/known data
    The Bloch-Gruneissen approximation for the resistance on a monovalent metal is

    [tex]\rho[/tex]=C(T[tex]^{5}[/tex]/[tex]\Theta[/tex][tex]^{6}[/tex])[tex]\int[/tex][tex]^{\Theta/T}_{0}[/tex][tex]\frac{x^{5}dx}{(e^{x}-1)(1-e^{-x})}[/tex]

    (a)For T->[tex]\infty[/tex], show that [tex]\rho[/tex]=(C/4)(T/[tex]\Theta^{2}[/tex])

    (b)For T->0, show that [tex]\rho[/tex]=5![tex]\zeta(5)[/tex]C[tex]\frac{T^{5}}{\Theta^{6}}[/tex]


    2. Relevant equations
    The section is on Bernoulli numbers and the Euler-Maclaurin Formula. Several definitions including x/(e^x-1)=sum->(Bn*x^n)/n!, Bernoulli Polynomials, Reimann-Zeta function, etc.


    3. The attempt at a solution
    For part (a) I see that as T->infinity the upper integration limit goes to zero, thus I may approximate the integrand giving (as the integrand) x^5/[(x+x^2/2!+x^3/3!+...)(-x+x^/2!-x^3/3!+...)]. Can I just multiply this out, simplify, and integrate term by term? If so, over what integration limits?

    For part (b) the upper limit goes to infinity so I'm guessing I need to do the integral by substituting some definition of the Bernoulli Numbers?...

    Help....

    PS - Sorry if the equations are unclear. I'm new to LaTex. Help with that would be appreciated too.
     
  2. jcsd
  3. Dec 6, 2008 #2

    Avodyne

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    Science Advisor

    Yep! (You have a sign wrong though ...)
    Same as before, you just think of the upper limit as a small parameter.
    Try expanding everything but the [itex]x^5[/itex] in powers of [itex]e^{-x}[/itex].

    As for LaTeX, just put the beginning and ending tex and /tex commands around the whole equation:

    [tex]\rho=C(T^{5}/\Theta^{6})\int^{\Theta/T}_{0}\frac{x^{5}dx}{(e^{x}-1)(1-e^{-x})}[/tex]
     
  4. Dec 6, 2008 #3
    I guess you mean the sign is wrong in the expansion of [tex]e^{-x}[/tex]? What is it's expansion?
     
  5. Dec 6, 2008 #4
    Isn't that what we did for part (a)?

    Thanks for the LaTex tip!
     
  6. Dec 6, 2008 #5

    Avodyne

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    I mean an expansion like

    [tex]{1\over 1-e^{-x}}=\sum_{n=0}^\infty (e^{-x})^n[/tex]
     
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