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The solution to the equation

$$Xe^X=K$$

is given by

$$X=W(K)$$

where $W$ is the Lambert function.

This idea was extended here to show that the solution to

$$\frac{1-e^X}{X}=K?$$

is given by

$$X = \frac 1K-W\left(\frac{e^{1/K}}K\right). $$

My question is, can we use a similar method to determine the solution to

$$\frac{\left(1 - e^{-X} \right)}{X} - e^X = K?$$

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David Simmons
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  • Interesting problem, for sure but I am afraid that, if solutions exist, only numerical methods could find them. – Claude Leibovici Oct 02 '14 at 10:02
  • I agree with @ClaudeLeibovici's comment - I don't see a solution in terms of $W$. At least, Mathematica couldn't find it. However, your situation is simpler in some ways. The left hand side of your equation is monotone decreasing as a real function and has a well-defined inverse. The properties of this inverse can be gleaned from the function itself and it's values can be easily computed numerically. Lambert's function, by contrast, is one branch of the inverse of a function that is not one-to-one. Of course, if you want to work in the complex domain, it's a bit different. – Mark McClure Oct 02 '14 at 10:28

1 Answers1

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For solving a given equation $F(x)=c$, $c$ constant, by isolating $x$ on one side of the equation, you can transform the equation if you can apply a suitable partial inverse $F^{-1}$ of the function $F$: $x=F^{-1}(c)$. In your equation

$$\frac{1-e^{-x}}{x}-e^{x}=k,$$

the left-hand side is an elementary function. It is an algebraic function that depends on the algebraic independent functions $x$, $e^{-x}$, $e^{x}$. It cannot be transformed into an algebraic equation of only one algebraic independent elementary function. According to a theorem of Ritt, such an elementary function cannot have an inverse that is also an elementary function. Therefore you cannot isolate $x$ by applying only elementary operations (elementary functions) to this equation.

We consider now the elementary functions and Lambert W function. To apply Lambert W, its inverse $xe^{x}$ is necessary. Together with the elementary functions, an equation

$$f_{1}(f_2(x)e^{f_2(x)})=c\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

with constant $c$ is needed where $f_1$ and $f_2$ are elementary functions with a suitable elementary partial inverse.

We therefore transform:

$$1-e^{-x}-xe^{x}=kx,$$

$$1-kx-e^{-x}-xe^{x}=0.$$

The left-hand side of the last equation is an algebraic function that depends on the algebraic independent functions $x$, $e^{-x}$, $xe^{x}$. Observing the form of equation (1), the elementary functions $f_1$ and $f_2$ cannot be transformed into algebraic functions of only one algebraic independent elementary function. Therefore you cannot isolate $xe^{x}$ by applying only elementary operations (elementary functions). Therefore you cannot apply Lambert W and only elementary operations (elementary functions) to isolate $x$ from your equation.

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