In this video I introduce the Lambert W Function, and work to demystify it, as well as work through a few simple algebraic examples using it. For videos on how this connects to Wien's law, visit: • Wien's Law
probably the sexiest explanation of the Lambert W function and its application. I don't usually comment, but you have done me such a service that I needed to comment!
So you still need a computer to evaluate the result and get a value? Every video I've seen on this function talks about the nice property, but conveniently neglects how one goes about evaluating the Lambert W function. So, we haven't really solved a problem here. We have just manipulated the equation and changed its form. It's not much different than saying "The solution to this equation is that answer that solves the equation."
In a sense I completely agree with you. Most special functions are defined to be the answer that solves some equation. But this doesn't mean that they aren't useful. Take a function like sine or cosine. One way of defining them is as the solution to the differential equation y''(x) = y(x). But in spite of this definition it is often more useful to work in terms of sine and cosine instead of leaving a problem otherwise unsimplified. In this case it's the ability to manipulate a problem more easily using the special function that makes it useful. In the case of the Lambert W function, it is nice to be able to write an exact function that is the inverse of xe^x, rather than working with an implicit form of the problem. Additionally, seeing the special function explicitly makes it easier to see and use the properties of the special function. I would also say that the process of evaluating the W function is comparable to evaluating a function like sine or cosine. What is sin(.2)? The way we find out is by plugging .2 into the Taylor series for sine and if we're working by hand then we evaluate a few terms and pay attention to the error. If we're on a computer then we just plug it in and get a very good answer. Likewise, there is a series for the Lambert W function (see my other videos), as well as other representations using integrals and continued fractions that one can use to get arbitrarily accurate values of the function. These can be approximated by hand, or you can just use a computer to get an exact answer. Thank you for the question. I hope my response makes some sense.
@@physicsandmathlectures3289 1/7 is the non-integer solution of 7x=1. sqrt(2) is the non-rational solution of x^2=2. sqrt(-1) is the non-"real" solution of x^2=-1. I'm wondering if there is a non-complex solution of tetrate( e^(-1*W(-1)) , -1/2) = sqrt(-1) that would allow us be analogous to i. Is there a name for this? P. S. I wrote e^(-1*W(-1)) there because it is one of two solutions to the equation ln(z)=z, where z is a complex number. I found a paper that says these are the only two values for which inverse tetration is not defined within the complex numbers.
Great explanation. I like how you make sense of it, rather than just throwing out a bunch of definitions. Also, it would be very useful if a link to the next video were in the description.
I am not a mathematician but I have watched a number of clips explaining the Lambert W Function. What I understood is that if one has an exponential function and one cannot solve it, one might resort to defining its inverse function. This means resorting to the Lambert (W) expression of that function, on the condition that the Lambert (W) is not defined for values smaller than minus (1/e). I hope I have expressed it properly. If not, please advise, correct so I can understand this properly.
I aproached this function in a very curious way. I was trying to determine in which intervals the function f(x)=xlnx-1 was positive or negative, thus having to calculate the value of x for f(x)=0. I tried to use the propierties of logarithms, but I was stuck in a cycle and didn’t manage to solve the equation. Then I tried to draw the graphic of the function and it seemed to cross with the x axis in somewhere near 1,73. I started substituting with the calculator values near that number and manage to get an aproximation of x. However, I still thought that there should be a way to get the exact value. I didn’t know what to do next, so, I asked, chat gpt, and it told me about the Lambert function. And that’s basically how I got to this video.
I don't think so. There are plenty of other representations using integrals or continued fractions, but I'm not aware of any elementary function representation.
@@physicsandmathlectures3289 Hm. Continued fractions (especially in Stern-Brocot type binary tree structures) are elementary in terms of proof theory. Not sure whether they can be called functions, though.
Longed to see an integer, decimal value, or complex number on the unit circle of Lambert W Function. So far am disappointed, and confused about the real use of Lambert W Function. Why no one has tabulated, to look up the values of Lambert W Function, live happily ever after? So far videos of Lambert W Function are hot air.
If the inverse of exponentiation gives us i=sqrt(-1), then what does the inverse of tetration give us? Find a complex z such that the inverse of tetration doesn't exist *in the complex numbers*, and define the inverse tetration of z by (-1/2) be a new number; call it j. Could j be a new dimension? Then we could parameterize a sphere with S(t)=S((x+iy)*jz)=tetrate(Z,e^(2pi*(1+i)*j^t)), where Z=inv.tetrate(i,e^(-1*W(-1))). I tried to work out Z (complex value) so that S(j)=j and S(S(j))= tetrate(j,j) = i. Also, the W-lambert function comes up in this value because I found a paper that says the only two complex values for which inverse tetration is not defined are the two complex solutions to ln(z)=z, which involve W-lambert. What is this called? I'm an amateur math studyist...I know it has to do with hyper operations... I'm trying to research the topic of extending the complex numbers to be closed under inverse exponentiation to define further dimensions/numbers. Is there a name/search term for this? If any of these ideas are useful, take it and run with it. Thanks y'all!
