The photon-number spectral integral of blackbody radiation is I = _0^ x^2 e^x - 1

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

The photon-number spectral integral of blackbody radiation is I=∫0∞x2ex−1 dx,I = \int_0^\infty \frac{x^2}{e^x - 1}\,\mathrm{d}x, which is known in closed form as I=2ζ(3)I = 2\zeta(3), with the Apéry constant ζ(3)=1.2020569\zeta(3) = 1.2020569. ((a)) Apply the substitution x=t/(1−t)x = t/(1-t), dx=dt/(1−t)2\mathrm{d}x = \mathrm{d}t/(1-t)^2, which maps an infinite range onto a finite one, and write II as an integral over t∈[0,1)t \in [0,1). ((b)) State, with a one-sentence reason, why a plain adaptive quadrature routine can now be applied safely to the transformed integral over [0,1][0,1] even though the original integrand decays only like x2e−xx^2 e^{-x} at infinity. ((c)) Give the numerical value of II to seven significant figures and compare with 2ζ(3)2\zeta(3).

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