The Fermi-Dirac-type integral I= _0^ x 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 Fermi-Dirac-type integral I=∫0∞xex+1 dxI=\int_0^\infty \frac{x}{e^x+1}\,\mathrm{d}x has the closed form I=π2/12I=\pi^2/12. A library adaptive quadrature routine cannot integrate directly to x=∞x=\infty. ((a)) Using the substitution x=t/(1−t)x=t/(1-t), which maps [0,∞)[0,\infty) to [0,1)[0,1), rewrite II as an integral over t∈[0,1)t\in[0,1), giving dx\mathrm{d}x in terms of dt\mathrm{d}t. ((b)) State, without computing further, why the transformed integrand is well behaved as t→1−t\to1^- even though the original domain is infinite there. ((c)) Evaluate II numerically (by this substitution, or by any equivalent standard method) to six significant figures and compare with π2/12\pi^2/12.

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