Editing Wien's displacement law (section)

==Derivation from Planck's law==
===Parameterization by wavelength===
Planck's law for the spectrum of black-body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is:
<math display="block">u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}.</math>

Differentiating <math>u(\lambda,T)</math> with respect to <math>\lambda</math> and setting the derivative equal to zero gives:
<math display="block">{ \partial u \over \partial \lambda } = 2 h c^2\left( {hc\over kT \lambda^7}{e^{h c/\lambda kT}\over \left(e^{h c/\lambda kT}-1\right)^2} - {1\over\lambda^6}{5\over e^{h c/\lambda kT}-1}\right) = 0,</math>
which can be simplified to give:
<math display="block">{hc\over\lambda kT } {e^{h c/\lambda kT}\over e^{h c/\lambda kT} -1} - 5 = 0. </math>

By defining:
<math display="block">x\equiv{hc\over\lambda kT },</math>
the equation becomes one in the single variable ''x'':
<math display="block">{x e^x \over e^x - 1}-5=0.</math>
which is equivalent to:
<math display="block">x = 5(1-e^{-x})\,.</math>

This equation is solved by
<math display="block"> x = 5+W_0(-5e^{-5}) </math>
where <math>W_0</math> is the principal branch of the [[Lambert W function|Lambert ''W'' function]], and gives {{nowrap|1=<math>x=</math> {{val|4.965114231744276303|end=...}}}}.<ref>{{Cite OEIS|1=A094090}}</ref>  Solving for the wavelength <math>\lambda</math> in millimetres, and using kelvins for the temperature yields:<ref>{{Cite journal|url=https://doi.org/10.1119/1.1466547|doi = 10.1119/1.1466547|title = Obtaining Wien's displacement law from Planck's law of radiation|year = 2002 | last1 = Das|first1 = Biman|journal = [[The Physics Teacher]]|volume = 40|issue = 3|pages = 148–149|bibcode = 2002PhTea..40..148D}}</ref><ref name=A081819 />
:{{nowrap|1=<math>\lambda_\mathrm{peak}=hc/xkT=\;</math>({{val|2.897771955185172661|end=...|u=mm⋅K}})<math>/T</math>.}}

===Parameterization by frequency===
Another common parameterization is by ''frequency''. The derivation yielding peak parameter value is similar, but starts with the form of Planck's law as a function of frequency <math>\nu</math>:
<math display="block">u_{\nu}(\nu,T) = {2 h \nu^3\over c^2}{1\over e^{h \nu/ kT}-1}.</math>

The preceding process using this equation yields:
<math display="block">-{h\nu\over kT }{e^{h\nu / kT}\over e^{h \nu /kT} -1} + 3 = 0. </math>
The net result is:
<math display="block">x = 3(1-e^{-x})\,.</math>
This is similarly solved with the Lambert ''W'' function:<ref>{{cite journal|doi=10.1021/ed400827f|title=A Specific Mathematical Form for Wien's Displacement Law as ''ν''<sub>max</sub>/''T'' = constant|year = 2014|last1 = Williams|first1 = Brian Wesley|journal = [[Journal of Chemical Education]]|volume = 91|issue = 5|page = 623|bibcode = 2014JChEd..91..623W|doi-access = free}}</ref>
<math display="block">
 x = 3 + W_0(-3e^{-3}) 
</math>
giving {{nowrap|1=<math>x</math> = {{val|2.821439372122078893|end=...}}}}.<ref name=A194567 />

Solving for <math>\nu</math> produces:<ref name=A357838 />
:{{nowrap|1=<math>\nu_\mathrm{peak}= xkT/h =</math> ({{val|0.05878925757646824946|end=...|u=THz⋅K<sup>−1</sup>}})<math>\cdot T</math>.}}

===Parameterization by the logarithm of wavelength or frequency===

Using the implicit equation <math>x = 4(1-e^{-x})</math> yields the peak in the spectral radiance density function expressed in the parameter radiance ''per proportional bandwidth''. (That is, the density of irradiance per frequency bandwidth proportional to the frequency itself, which can be calculated by considering infinitesimal intervals of <math>\ln\nu</math> (or equivalently <math>\ln\lambda</math>) rather of frequency itself.) This is perhaps a more intuitive way of presenting "wavelength of peak emission". That yields {{nowrap|1=<math>x</math> = {{val|3.920690394872886343|end=...}}}}.<ref>{{Cite OEIS|1=A256501}}</ref>