# About the solvability of non-analytical equations with new “helper” functions

VtC reviewers: please drop a comment, what "context or other details" are missing.

I am thinking on to solve such equations like $e^x+cx+d=0$. These are generally not analytical, which essentially means that we can't do with them anything. There is no closed formula what could express the result.

However, it is not the first case as we faced similar problems. The ancient greeks had a quite similar problem about the irrationals. But we don't have a finite expression of, for example, $\sin22^\circ$, too.

The solution was that new functions were invented, their details, identities were discovered. For example, we still don't have a closed expression for $\sin 22^\circ$, but trigonometry is high school math today, and most practical trigonometric problems can be easily solved.

For example, what if we define a function

$\kappa(c)$ on the way, that $\kappa(c) := x|e^x+cx=0$ ?

I suspect, this $\kappa(c)$ would have identities, it would have Taylor-series, and using these we could get the solution for not only the equations like $e^x+cx=0$, but for a far more wider class of equations?

Do such functions exist? More specifically, what is the case for the named problem ("roots" of $e^x+cx+d$)?

• BTW, $\sin(22^\circ)$ has "closed form" expressions involving radicals (not real radicals though). – Robert Israel Jun 9 '18 at 1:09

For the specific question... Solution $x$ of $e^x+cx+d=0$ is $$x = -\frac{d}{c}-W\left(\frac{\exp(-d/c)}{c}\right)$$ where $W$ is the Lambert W function.

• Wow! So, these functions exist! Thanks! :-) – peterh Jun 9 '18 at 0:33