Articles of lambert w

Two exponential terms equation solution

Let $A_i$ and $B_i$ denote constants, I know this equation $$A_1 \exp(B_1x) + A_2x + 1 = 0$$ can be solved using lambert W function. But can I get a general solution of this equation? $$A_1 \exp(B_1x) + A_2 \exp(B_2x) + A_3x + 1 = 0.$$ I searched a lot but did not get an […]

A strange identity related to the imaginary part of the Lambert-W function

Working on a problem in QFT, i was stumbeling about some expressions containing the Lambert-$W$ function. Playing around, i discovered experimentally that the following statement seems to be true $$ \Im (W_0(-x))=-\Im (W_{-1}(-x))\,\,\, \text{if} \,\,\, x>1/e $$ For large $x$ we can write $W_0(-x)\approx\log(-x)$ and $W_{-1}(-x)\approx\log(-x)-2 \pi i$ choosing now $\log(-x)=\pi i+\log(x)$ gives the desired […]

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ $\Rightarrow -2x=W(\dfrac{-1}{3})$ $\therefore x=\dfrac{-1}{2} W(\dfrac{-1}{3})$ But when i went to WolframAlpha, it showed the same result but in the graph: WolframAlpha Graph http://www5b.wolframalpha.com/Calculate/MSP/MSP132207dchg91df64hcb0000351g9904e7fi986a?MSPStoreType=image/gif&s=61&w=349.&h=185.&cdf=Coordinates&cdf=Tooltips The curves intersect at a point… And hence there is a second […]

Graph of the function $x^y = y^x$, and $e$ (Euler's number).

Earlier, I was using the Desmos Graphing Calculator, and I wanted to remind myself of what the graph of the function $x^y = y^x$ looked like. If you have never seen what it looks like before, it is similar to the shape of the Greek letter, psi (ψ); composed of two graphs: $y = x$ […]

New Elementary Function?

In the February 2000 issue of FOCUS magazine, a short article suggests that the Lambert W function could be introduced into curriculum as a new elementary function saying: “… a case can be made for according it equal respect with the traditional transcendentals of calculus.” As the inverse of $xe^x$, Lambert W is easy to […]

Are those two numbers transcendental?

Suppose, $u$ solves the equation $$u^u=\pi$$ and $v$ solves the equation $$v\cdot e^v=\pi$$ So, we have $u=e^{W(\ln(\pi))}$ and $v=W(\pi)$. $u$ and $v$ should be the real solutions (in this case, they are unique). If someone can prove that $u$ and $v$ are transcendental, the next step would be to show that every solution is transcendental, […]

Another interesting integral related to the Omega constant

Another interesting integral related to the Omega constant is the following $$\int^\infty_0 \frac{1 + 2\cos x + x \sin x}{1 + 2x \sin x + x^2} dx = \frac{\pi}{1 + \Omega}.$$ Here $\Omega = {\rm W}_0(1) = 0.56714329\ldots$ is the Omega constant while ${\rm W}_0(x)$ is the principal branch of the Lambert W function. A […]

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, primarily by exploring various manipulations using logarithms and polylogarithms but have gotten nowhere. Although it is simple enough to show that $y(\sqrt{2})>y(1)$ and if $y'(x)>0$ for some […]

Lambert W function aproximation

I noticed that $\ln(x)-\ln(\ln(x))$ seems to by asymptotic with the Lambert W Function. Is this true?

Solving equation involving self-exponentiation

How do I solve the equation $\displaystyle x=ay^2(by)^{\frac 1y}$ for $y$, where $a$ and $b$ are constants? I’ve been trying to manipulate this into a form on which I can use the Lambert W function, but I don’t know whether this is possible or if so how to do it. Sorry about the title, I […]