What is the Lambert W Function (Introduction )/ Part 1

In this Video I am going to explain you what is the Lambert W function. Since this is an introduction to the Lambert W Function I will begin with a historical view of this function.
The Lambert W function is a mathematical function with a long history, but which was named and rigorously defined relatively recently. It is closely related to the logarithmic function and arises from many models in the natural sciences, including a surprising number of problems in ecology and evolution.
In this Video I am going to show you how to graph the Lambert W Function if you know the original function.

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[andrefrancis]

Yes, I agree with Calvin Jackson. Presenters introduce and use Lambert W without really explaining it in a way that is clear. You have done this so well. Congratulations! I am a medical statistician (retired from operational work but still researching) and interested in how W might be relevant in transformations involving clinical variables.

[calvinjackson]

You are the only one who has made this function perfectly clear to me. I want to thank you so much. Thanks for explaining it so clearly!

[tharock]

So I watched one video where the Lambert function was used, and now I see them all the time, but it's just this random function everyone uses along with some algebraic trickery to answer a hard exponential equation. Thanks for explaining what it actually is beyond just something you use for answering Math Olympiad problems.

[peternolan]

Hello Ivana,

I will be 68 in July. This video has helped me understand what The Lambert W Function is. I have watched other videos about The Lambert W Function and I didn't know how confused I was till I saw your video and the graphs you have drawn of:

y = xexpx

and The Lambert W Function as the inverse of:

y = xexpx

I didn't know how to calculate the inverse of a function by replacing the x with a y and the y with an x.
I wrote down this differential equation:

dv/dt = a((1 -2v/c)/(1 -v/c))

where v is the speed and a is the acceleration and c is the speed of light and when I used Wolfram's Mathematica to solve this first order differential equation the solution is:

In[1] := DSolve[{EQN, v[0] == 0}, v[t], {t}]
Out[1] = {{v[t] -> 1/2 (c - c ProductLog[E^(1 - (4 a t)/c)])}}

The ProductLog is The Lambert W Function. I didn't know anything about The Lambert W Function before.

All the best and many thanks,

Peter Nolan. Ph.D.(physics). Dublin. Ireland.

[yarabamba]

I revirewed many videos on the topic. This is the clearest, Kudos.

[rah]

Thanks for this great playlist! I only recently found your channel which is very good. Please start posting again!

[HAL-ojjb]

Nice video, really looking forward to that series! I see that function so often on math videos without really explaining what it really is and I'm stoked to learn something new ^^

[Tomorrow]

I luv it when German women speak English ❤️

[Shihabuddin]

Thank u so much❤❤. I am from Bangladesh

[segayanmx]

Thank you for your explanation! Could you explain us Lambert-Tsallies method ?

[drdiegocolombo]

Hi Ivana and thanks for your videos. It's a W Lambert function Tshirt that?😀

[DanBurgaud]

discussing Lambert W function and wearing WW shirt...

[incomestockinvesting]

Thanks for the history lesson on some math. Very cool. ☺️

[keithturvey]

Lambert is well-known amongst students of physics for his work on the luminance of surfaces that emit light; a so-called Lambertian surface has the same luminance when viewed from any direction. As a former, but now retired, lecturer in physics I was familiar with that aspect of his work but only came across the mathematical function named after him within the last few days. By referring to Wikipedia I found that in both cases it is the same Lambert, i.e Johann Heinrich Lambert. This prompts me to ask: Did he introduce the function that was later named after him because to him it had some particular application in physics or perhaps even in the narrower field of optics?

[__hannibaalbarca__]

This is a limit of infinite iterated exponential. After analytical extend over complex.

[SPV]

@intellecta2686
Thanks for the clear and good explanation.

Just one thing, at 5:36 you say " ... and our e is approximately two-point-seventy-one ...".

The individual digits after the decimal point should be stated individually and not collectively. For example, when saying a number like 1.75, it should be said "one-point-seven-five" and not "one-point-seventy-five".

So, e is approximately two-point-seven-one and not two-point-seventy-one.

[stevenparker]

hi, I'm still not clear on how to use this function, can you point me to some real world examples or 'simple' problems to solve so I can get a better understanding of the function?

[mouradbelkas]

You really need to review your math skills and understand what is the Lambert W function. Go to internet and learn it well.. Thank you for trying.

[ravivaradhan]

y = xe^x attains its minimum value of (-1/e) at x=-1

[AubreyForever]

It looks like the function is a reflection rather than an inverse.