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Find the Intersection of Einstein's E4 Function With Hyperbola
Reduce can be used directly to compute the intersection of Einstein's E4 function with the unit hyperbola.
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Click for copyable input
Subscript[E, 4] = x/(Exp[x] - 1) - Log[1 - Exp[-x]];

intp = Reduce[Subscript[E, 4] == 1/x, x, Reals]
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In[2]:=

Click for copyable input
Plot[{Subscript[E, 4], 1/x}, {x, 0, 3}, 

 Epilog -> {Red, PointSize[0.025], 

   Point[{x, Subscript[E, 4]}] /. {ToRules[intp]}}]
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