Wikipedia:Reference desk/Archives/Science/2024 August 23

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August 23

Planck's law 1901 article and reverse function

At the end of the chapter "...Wien's displacement...", after equation (8), Max Planck gives the formula:
$U=\nu \cdot f_{1}({\frac {\theta }{\nu }})$
Then a new formula:
$\theta =\nu \cdot f_{2}({\frac {U}{\nu }})$
Ok, but the second formula that follows from it is incomprehensible to me:
${\frac {1}{\theta }}={\frac {1}{\nu }}\cdot f_{3}({\frac {U}{\nu }})$
One has:
$f_{3}({\frac {U}{\nu }})={\frac {1}{f_{2}({\frac {U}{\nu }})}}???$
Any idea ?
Malypaet (talk) 12:58, 23 August 2024 (UTC)

So $\theta =\nu \cdot f_{2}({\frac {U}{\nu }})$ and ${\frac {1}{\theta }}={\frac {1}{\nu }}\cdot {\frac {1}{f_{2}({\frac {U}{\nu }})}}$, it's just taking the reciprocal of both sides. But I don't do physics so I'm probably missing the point.  Card Zero  (talk) 13:38, 23 August 2024 (UTC)

My question is:
on what logic can we write:
$f_{3}({\frac {U}{\nu }})={\frac {1}{f_{2}({\frac {U}{\nu }})}}???$
Malypaet (talk) 14:14, 23 August 2024 (UTC)

Still unsure if I'm really helping, but so long as I don't have to know anything about black-body radiation or whatever,

If $a=b\cdot c$ then ${\frac {1}{a}}={\frac {1}{b}}\cdot {\frac {1}{c}}$, so

If $a=b\cdot {f_{2}(x)}$ then ${\frac {1}{a}}={\frac {1}{b}}\cdot {\frac {1}{f_{2}(x)}}$, and

If ${\frac {1}{a}}={\frac {1}{b}}\cdot {f_{3}(x)}$ then

${f_{3}(x)}={\frac {1}{f_{2}(x)}}.$ But I'm just filling space until somebody comes along who knows what you were getting at.  Card Zero  (talk) 15:08, 23 August 2024 (UTC)

Nobody knows what Malypaet is trying to get at... The answer here, I guess, is simply that $f_{3}$ is a new name for $1/f_{2}$, nothing more, nothing less. Planck doesn't know what $f_{2}$ looks like (all he knows is that its argument is $U/\nu$), and he doesn't know what $f_{3}$ looks like (all he knows is that, because $f_{3}=1/f_{2}$, it is also a function of $U/\nu$). --Wrongfilter (talk) 15:46, 23 August 2024 (UTC)

Written like that, we can admit it. In his combinatorial demonstration we find this analogy of functions between logarithms and exponentials. But he does not write it.
Thank you. Malypaet (talk) 18:59, 23 August 2024 (UTC)