Equation 87 · No Particle Without a Cosigner

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u^\ast \approx 0.25363

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u^\ast

Symbol u^ast

$u^a$ st is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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≈

Approximately equal to; the equality is not exact.

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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\, $u^2$ /( $e^{2\pi u}$ -1) means solving $e^{2\pi u}(1-\pi u)$ =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 $u^\ast$ $\approx$ 0.25363 , at which $u^\ast$ g( $u^\ast$ ) $\approx$ 0.3937 . The leading-order sensitivity is…

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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\, $u^2$ /( $e^{2\pi u}$ -1) means solving $e^{2\pi u}(1-\pi u)$ =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 $u^\ast$ $\approx$ 0.25363 , at which $u^\ast$ g( $u^\ast$ ) $\approx$ 0.3937 . The leading-order sensitivity is therefore

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Sources cited in the article section

[9] Asymptotic States of Accelerated Detectors and Universality of the Unruh Effect ↗

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