dcasimir

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Prove that  ((sin(x)/x)^2, x=-infinity..infinity) = Pi/2

A given hint is (sin(x))^2 = (1 - cos(2*x))/2

 

Thanks,

rho[nu] = 4*Pi/(h^3)*Int(x^3/(exp(x/(k*T))+1), x = 0..infinity)

How can I show that the above integral equals 75*Pi^5*(k*T)^4/(30*h^3)

 

This is from Arfken's Math Methods for Physicists in Chapter 8.2, on the Digamma and Polygamma functions.

Please help,

getting frustrated,

 

very respectfully,

 

 

This problem (problem # 30) from Goldstein's "Classical Mechanics" specifically asks one to use Maple. The problem is as follows. Using Maple or Mathematica or a similar program calculate the Einstein field equations for spherical coordinates assuming T[mu,nu] = 0 everywhere except possibly for r = 0, where the coordinate system is undefined. The most general spherical static metric corresponds to an interval given by (ds)^2 =e^(nu(r))*(c^2)*(dt)^2 - e^(lambda(r))*(dr^2) - r^2*((dtheta^2) + sin^2(theta)*(dphi^2)), where r, theta, and phi correspond to the usual three-dimensional spherical coordinates.
How do circles centered on the origin in the z-plane transform for ? (a)w[1](z) = z + (1/z)
Show that the substitution x -> (1-x)/2
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