An Introduction to Thermal Physics · Problem B.14

Systems of Interacting Particles — Daniel Schroeder

Physics — Statistical Mechanics · hypersphere-area

Problem

The proof of formula B.28 is by induction. (a) Check formula B.28 for n=0n = 0 and for n=1n = 1. (b) Show that ∫0π(sin⁡θ)n dθ=(n−1n)∫0π(sin⁡θ)n−2 dθ.\int_0^{\pi} (\sin\theta)^n\, d\theta = \left(\frac{n-1}{n}\right) \int_0^{\pi} (\sin\theta)^{n-2}\, d\theta. (Hint: First write (sin⁡θ)n(\sin\theta)^n as (sin⁡θ)n−2(1−cos⁡2θ)(\sin\theta)^{n-2}(1 - \cos^2\theta). Integrate the second term by parts, differentiating one factor of cos⁡θ\cos\theta and integrating everything else.) (c) Use the results of parts (a) and (b) to prove formula B.28 by induction.

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