Unable to solve equation explicitely

asked 2023-10-03 04:05:31 +0200

Venator gravatar image

updated 2023-10-03 08:30:47 +0200

Emmanuel Charpentier gravatar image

I have the following equation:

U=(8*pi*h*f^3)/(c^3)/(exp((h*f)/(k*T))-1),

and I would like to solve it's derivative with respect to f equal to zero for f. I know this equation can be explicitly solved for f with the LambertW function, however sage only provides an implicit solution, or if I specify it must be explicit, it only gives the trivial solution. Is there any way to get it to use the LambertW function when using solve?

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Edited for legibility : to quote code, indent it by four spaces...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2023-10-03 08:56:05 +0200 )edit

For the benefit of future commenters/answerers, a bit of Sage code :

sage: var("U, h, f, c, k, T")
(U, h, f, c, k, T)
sage: U=(8*pi*h*f^3)/(c^3)/(exp((h*f)/(k*T))-1)

Sage gives indeed an implicit answer :

sage: solve(diff(U, f)==0, f)
[f == 0, f == 3*(T*k*e^(f*h/(T*k)) - T*k)*e^(-f*h/(T*k))/h]

Neither Sympy, Giac nor Fricas can solve this one.Mathematica gives an explicit answer using Lambert's W :

sage: mathematica.Reduce(diff(U, f)==0, f)
Element[C[1], Integers] && h != 0 && 
 c*(-1 + E^(3 + ProductLog[C[1], -3/E^3]))*k*T != 0 && 
 f == (k*T*(3 + ProductLog[C[1], -3/E^3]))/h

ProductLog being indeed Marhematica's name for Lambert's W.

HTH,

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2023-10-03 09:29:25 +0200 )edit

It is easy to solve the equation manually using the definition of Lambert W function.

Max Alekseyev gravatar imageMax Alekseyev ( 2023-10-03 13:23:11 +0200 )edit
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