- #1
kde2520
- 16
- 0
Homework Statement
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]
Homework 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.
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.