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December 14th, 2016, 07:37 AM   #1
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Solving e^x - x = a

What's the mechanism for solving an equation of this kind?

Last edited by skipjack; December 17th, 2016 at 04:37 AM.
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December 14th, 2016, 08:00 AM   #2
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You're going to have to solve this numerically.

There is a closed form solution

$x = -W\left(-e^{-a}\right)-a$

where $W(x)$ is the Lambert W function

but $W(x)$ doesn't generally show up on your calculator.

There are no lack of numerical algorithms to solve equations like this.

Newton's method is a good place to start.
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December 14th, 2016, 08:13 AM   #3
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Thanks a lot! I'll start with learning about the Lambert function and the Newton method.

Last edited by skipjack; December 14th, 2016 at 11:35 AM.
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December 16th, 2016, 05:59 PM   #4
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This is a trascendental equation, where the function on the left equals the function on the right:

\begin{eqnarray}
e^{x}=x+a
\end{eqnarray}

The solution is calculated graphically, by plotting both functions as a function of $x$. All possible solutions to the equation are then those values of $x$ where both graphs cross.
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December 17th, 2016, 04:48 AM   #5
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If $a$ < 1, there is no real solution. If $a$ = 1, $x$ = 0 is the only real solution. If $a$ > 1, there are two real solutions.
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