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e2πu(1−πu)=1e^{2\pi u}(1-\pi u)=1

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This is the same trick used to derive Wien’s displacement law: because u\,g(u) rises from zero, passes through a single interior maximum, and falls back to zero, the integral of the absolute value of its derivative over all u is exactly twice that maximum, with no need to evaluate the integral directly. Finding the maximum of u\,g(u) = 24\,u2u^2/(e2πue^{2\pi u}-1) means solving e2πu(1−πu)e^{2\pi u}(1-\pi u)=1 , the same class of transcendental equation that fixes the peak of a Planck spectrum weighted by an extra power of frequency. Solved by Newton’s method to five-figure precision, the nontrivial root is u∗u^\ast ≈\approx 0.25363 , at which u∗u^\ast g(u∗u^\ast) ≈\approx 0.3937 . The leading-order sensitivity is…

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e2πue^{2\pi u}

Symbol e^2pi u

e2e^2pi u is part of the quantity the equation computes from the expression on the right.

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e2πu(1−πu)=1e^{2\pi u}(1-\pi u)=1

Equation 86 · Evolutionary Physics

No Particle Without a Cosigner

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

This is the same trick used to derive Wien’s displacement law: because u\,g(u) rises from zero, passes through a single interior maximum, and falls back to zero, the integral of the absolute value of its derivative over all u is exactly twice that maximum, with no need to evaluate the integral directly. Finding the maximum of u\,g(u) = 24\,u2u^2/(e2πue^{2\pi u}-1) means solving e2πu(1−πu)e^{2\pi u}(1-\pi u)=1 , the same class of transcendental equation that fixes the peak of a Planck spectrum weighted by an extra power of frequency. Solved by Newton’s method to five-figure precision, the nontrivial root is u∗u^\ast ≈\approx 0.25363 , at which u∗u^\ast g(u∗u^\ast) ≈\approx 0.3937 . The leading-order sensitivity is…

Meanings in this article

  • uu: exactly twice that maximum.
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