(PCb23) What is the Lambert-W Function?

In this video, we introduce the Lambert-W function and its unique property for which it inherits from its definition. We then use the Lambert-W function to solve two types of (transcendental) algebraic equations that one would not be able to solve with the other elementary functions.

--The Let's Learn, Nemo Community--*

#LetsLearnNemo #Mathematics #PreCalculus

[laman]

I have watched a number of clips on YT explaining the Lambert W Function and this one really explains it easy: when to apply it, its use, and properties. Thank You.

[tanelkagan]

I understand the "what?" and I understand the "how?", but I still don't understand the "why?".

What we've done is construct a function which from I can see is the inverse of x.e^x, and ok, that's fine. But why is x.e^x special, and why is creating an inverse function for that x.e^x useful?

I can see how you talk about "solving" equations using the function, but it would seem to me that all that is happening is that the equation is changing form, and we're actually introducing something extra into it, the W.

I understand the comparisons with square roots and with logs, but in the end those functions can be related back to numbers so that we are closer to evaluating or enumerating our original unknown quantities. We might not know exactly what the square root of 52 is, but we do know that it is just a number, and we know that that number is something slightly larger than 7. We can at least give some "meaning" to our answer and we would certainly want that in any practical application.

I am not clear on how the W function adds meaning - it seems to be a simplification of sorts, but just gives me an answer in terms of something new, that I didn't want in the first place. What would stop me creating a "WW" function and letting it be the inverse of x^2.e^x? Is this something that would or be universally recognised, or would everyone simply say "so what?"

Perhaps there are answers to these questions that come further "down the line", but at the moment I don't see what we gain by using this function. It seems to be like asking "What is black?", the answer coming back as "The opposite of white", then asking "Well, then, what is white?", with the answer coming back "The opposite of black".

Your further help would be appreciated!

[poijmc]

Great video! I was searching for the equation for this function... Now I know it's only the definition ;(

[theimmux]

Solve x·e^x=7.

"Oh, the solution is just simply the number that equals 7 when multiplied by e to the power of the number."

[SuryaBudimansyah]

04:20 "....but we will not talk about them here, ...."
Hey where I can find that 'numerical methods to calculate' the Lambert W function?

[carokann]

Hi, I would like to ask a question, How is the product log function of a number computed? I know of some methods for the real numbers like newtons method which makes sense but what about other branches of this function, like w with subscript 6 or -18 or whatever, what do those even mean? I know that it’s a multivalued function it still doesn’t make sense to me mind explaining? Great video btw

[pocojoyo]

thanks ! what software and hardware do you use for your lessons ?

[niubiteebee]

can you define w in terms of a simpler way of writing it? Because I would like to use my casio fx100au plus. I am from Australia doing Grade 11 Advanced mathematics (pretty nomral level) and I need help with trigonometry, but i really like the x^x=some number equations and I have a particular interest in lambert w function and newtons method.
So to summarize, I would like to know how to use the lambert w function on my casio calculator, or is it too advanced for my calculator that I need to go on wolfram alpha?
Great video
Thanks

[wizard]

What about x^x^x=-1? (Or just X hyper-4 3 = -1)

[wizard]

I notice that super square root of y = e^W(ln(y)). Can I use the super square root function instead of the W-lambert function in everything?