Transcendental Equation

After playing around with the Planck radiation law for the spectral energy density, (differentiation and setting equal to 0 to find the maximum wavelength $x$) I'm stuck on how to deal with what I'm left with.

I'm having difficulty trying to solve and plot the roots of this equation. I know it's a transcendental equation and will have to solved numerically though I'm at a loss how to do this on programs such as Wolfram Mathematica 7 or MATLAB.

Could not get this to work: $$x=\frac{a \exp(a/x)}{-5 (\exp(a/x)-1)}$$

where $a$ is a constant

• Please include your equation as code that can be copy and pasted - also it would be nice if you said what you had tried so that people don't waste time trying the same things. – Simon Woods Apr 20 '15 at 20:24
• I'm not sure, but I don't believe this can be solved on Mathematica 7. – Michael E2 Apr 21 '15 at 1:28

A plot like that in the reference provided in the Question can be obtained as follows. Solve the given equation (with x normalized to a for convenience and without loss of generality):

Reduce[x == Exp[1/x]/(5 - 5 Exp[1/x]), x]
(* Element[C[1], Integers] && -1 + E^x^(-1) != 0 &&
-5 + ProductLog[C[1], 5*E^5] != 0 && x == (-5 + ProductLog[C[1], 5*E^5])^(-1) *)


and pick out the solution,

x = %[[4, 2]] /. C[1] -> i
(* (-5 + ProductLog[i, 5*E^5])^(-1) *)


Plotting is straightforward.

ListPlot[Quiet @ Table[N[{Re[x], Im[x]}], {i, -20, 20}], PlotRange -> All]


• In v10.1 {Re[x], Im[x]} can also be written ReIm[x] – Bob Hanlon Apr 21 '15 at 0:07
• @BobHanlon Something I did not know. Thanks! Strangely, Table[N[ReIm[z]], {i, 1000000}]; is slightly slower than Table[N[{Re[z], Im[z]}], {i, 1000000}]; measured with AbsoluteTiming - 2.5 vs 2.2 sec on my PC.. I would have expected it to be as much as a factor of two faster – bbgodfrey Apr 21 '15 at 21:37