Equation 75 · No Particle Without a Cosigner

What does this equation mean?

\int_0^\infty x(e^x-1)^{-1}dx=\pi^2/6

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationspi^2/6

Result or conditionint_0^infty x(e^x-1)^-1dx

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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x

Symbol x

x is part of the quantity the equation computes from the expression on the right.

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e^x

Symbol e^x

$e^x$ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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d

Symbol d

d is part of the quantity the equation computes from the expression on the right.

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\pi^2

Symbol pi^2

p $i^2$ is an input to the expression that computes the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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0

Starting index or lower bound: 0

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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\infty

Ending index or upper bound: infty

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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How to interpret it

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What the article says around this equation

Write the normalized accelerated spectrum in terms of the dimensionless ratio u $\equiv$ c $\Omega$ /a , using the exact Bose-Einstein shape derived above. Carrying out the normalization integral, $\int_0^\infty$ $\Omega$ \,( $e^{2\pi c\Omega/a}$ -1)^{-1}\,d $\Omega$ = $a^2$ /(24 $c^2$ ) , using the standard result $\int_0^\infty$ x( $e^x$ -1)^{-1}dx= $\pi^2$ /6 , gives the scale-family form

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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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