In the nonrelativistic regime,
The assumption puts the Fermi-Dirac distribution in its Maxwell-Boltzmann limit, so
Therefore the nonrelativistic Maxwell--Boltzmann number density is
For , chemical equilibrium and the vanishing photon chemical potential imply
Applying the number-density formula to each massive species and eliminating the chemical potentials gives
Use charge neutrality, , the mass approximation , and the convention in the question that suppresses the order-one internal-degeneracy ratio . With the Hydrogen binding energy
we obtain the Saha ionization equation
The assumptions are thermal equilibrium and chemical equilibrium before cosmological recombination, a nonrelativistic nondegenerate gas, zero photon chemical potential, charge neutrality, negligible proton-electron mass correction in the translational prefactor, and the stated degeneracy-factor approximation.
For
a finite point is an ordinary point exactly when and are holomorphic there. It is a regular singular point exactly when
are holomorphic at .
Put and . Direct differentiation gives
so the transformed equation is
Consequently is ordinary precisely when
are holomorphic at . It is a regular singular point at infinity precisely when and are holomorphic functions of near infinity.
If zero and infinity are regular singular and every nonzero finite point is ordinary, the Laurent series of and can contain only the terms compatible with both endpoint bounds. Hence
for constants . The equation is a Cauchy-Euler differential equation. Its indicial equation is
For distinct roots , the general solution on a domain with a chosen logarithm branch is
For a repeated root , the general solution of an Euler-Cauchy equation is
Finally require infinity to be ordinary. In the transformed equation its coefficients become
Both are holomorphic at zero exactly when and . Thus the further restriction is
With , the equilibria satisfy
Besides , the two positive equilibria are
where . Since
the nonzero equilibrium relation gives
The fixed point stability for an autonomous differential equation therefore shows that and are stable, while is unstable. The graph of starts at zero with negative slope, crosses upward at , crosses downward at , and tends to as .
For a constant input , write the equilibrium equation as
The low stable equilibrium and the intervening unstable equilibrium coalesce in a saddle-node bifurcation. More precisely, let be the first positive solution of
and define
Equivalently, and , with on the low-concentration branch. If , that branch no longer exists. Holding the input long enough carries the trajectory into the basin of attraction of the high state. When the input returns to zero, the concentration converges to
This is the saturating autocatalytic switch.
For , the threshold occurs at , so
The approximate double-root conditions are
Thus and
Hence the constant in is , as recorded by the strong-autocatalysis switching threshold.
A one-parameter exponential family with natural statistic has density or probability mass function
where is the natural parameter of an exponential family and is the cumulant function of an exponential family. Its mean parameter is
The exponential-family deviance from to is twice the Kullback-Leibler divergence:
The carrier cancels from the likelihood ratio, so, with ,
For the Poisson distribution with mean ,
Therefore
Writing and applying the Taylor series of gives
so the second-order approximation is

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
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    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
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