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Say I have an equation that I can solve in $x$ as follows:

$$ x = LambertW_{-1}(y)$$

Where LambertW is the product-log function.

Can I say I have an explicit solution for $x$? It looks like that, but I'm not sure, given that I used the product-log function to solve for $ye^y$ - something which doesn't have an explicit solution in my books.

Is using the product-log function just cheating? Is it just a list of numerical values saved somewhere? I know that the derivatives are well-defined in term of other functions, but that's not really sufficient.

Define explicit solution

How about we call it closed-form solution then. To quote Wolfram Alpha:

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.

Does Lambert-W belong into this "generally-accepted" set?

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    $\begingroup$ I do not think that this is cheating. It is the solution of $y=x e^x$ and Lambert is a special (beautiful) function. $\endgroup$ – Claude Leibovici Mar 8 '16 at 11:43
  • $\begingroup$ @Claude Leibovici : I fully agree with you. $\endgroup$ – JJacquelin Mar 9 '16 at 12:05
  • $\begingroup$ I'd say the Lambert-W is still working its way into becoming "generally-accepted" and I believe there are some articles concerning this subject. Due to the applications of this function, it may very well become taught in schools. $\endgroup$ – Simply Beautiful Art Mar 9 '16 at 12:38
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The closed form of a special function is itself.

The real question is : Is the LambertW function a special function ?

To answer, we have to look on the list of standard functions. The hitch is that there is no definitive or exhaustive list of such functions.

What is a special function ?

It's a function (usually named after an early investigator of its properties) having particular use in mathematical physics or some other branch of mathematics. From : http://mathworld.wolfram.com/SpecialFunction.html

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications : https://en.wikipedia.org/wiki/List_of_mathematical_functions

The LambertW function appears in the mathematical litterature for a long time, is well-known and is implemented in most of the mathematical softwares. It this sens, we can say that the LambertW function has reached the honorific rank of special function.

Hense, we are not cheating when we express a result of calculus in which the LambertW function is involved.

An article for the general public on this subject : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales

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First of all, an explicit solution is not so well defined from you, so I will try to do my best to satisfy your question.

Is using the product-log function just cheating?

Maybe, but if it is cheating, aren't logarithms cheating? Isn't the entire base of trigonometry just functions we made up for convenience? In essence, I could 'invent' a function $f(x)$ that was the inverse of $g(x)=x+\sin(x)$, which naturally doesn't exist using a finite amount of known functions. If people found it useful enough, then maybe this function will get a name and people will use it.

The Lambert W function was just as similar. Someone (Lambert W) had a problem involving exponential functions and he couldn't solve it using known functions. So, Lambert W simply 'made' a new function, the Lambert W/product log function. Later, people saw that it was useful and had applications, so now this function is more 'official' and more accepted.

Is it just a list of numerical values saved somewhere?

No, you can actually approximate the product log using linear approximation, for example. I would think saving some numerical values, including complex values, would be too tedious. You should be able to find many online calculators that can solve for complex inputs and even a choice of which branch of the product log you want.

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  • $\begingroup$ I've tried to improve on the definition of explicit solution. Does that make your answer more specific? $\endgroup$ – FooBar Mar 9 '16 at 9:34

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