Transcendental Equation, and Lambert's W-Function

Discussion in 'Physics & Math' started by Facial, Aug 3, 2014.

  1. Facial Valued Senior Member

    As a good exercise, it should be noted that the 2-parameter Weibull distribution probability density function (PDF)

    \( \rm PDF(\it x\rm) = \frac{\rm\alpha \it x^{\rm\alpha-1}}{\rm\beta^{\rm\alpha}}\left.\right. e^{\(\frac{x}{\beta}\)^\alpha} \)

    has a closed-form solution for x in terms of the product log function, also known as Lambert's W-function. I obtained the solution by hand, and it should also be possible with computer algebra systems. Finding the expression for x should be a good warm-up to the question I would like to answer, which follows next.

    With the product log, currently I am trying to determine whether the equation

    \( \frac{1}{2z}\sqrt{z^2 \:+ \:1}\: +\: \frac{z}{2}\rm arcsinh(\it \frac{1}{z})\left.\right. -\left.\right. 1 = \rm C\it\sqrt{z^2 + 1}\)

    has a closed-form solution using the product log function. I haven't succeeded yet. It may be good to recall the identity

    \( \rm arcsinh(\it x\rm) = \rm ln(\it x + sqrt{x^2 + 1}) \) as a reference.

    I only ask because I have been trying to solve this for several months now to no avail, and do not have higher knowledge of theorems (i.e., existence of solutions) in math. If it's possible, it definitely should be a decent challenge.
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  3. rpenner Fully Wired Valued Senior Member

    You appear to be missing a minus sign, at least in regard to the sources I checked.

    Do you mean a transform from a uniform distribution over [0,1) to a random variable with a Weibull distribution? That's not very hard.

    There's not the slightest indication that such a thing would be true.
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  5. Facial Valued Senior Member

    Ah yes, that's right, I forgot the minus sign. The correct expression for the 2-parameter Weibull distribution is

    \( \rm PDF(\it x) = \frac{\alpha x^{\alpha - 1}}{\beta^\alpha} e^{-\left(\frac{x}{\beta}\right)^\alpha} \)

    My knowledge of math is a little limited. Are there references you can point me towards?

    If it's impossible, then I suspect it may be possible by defining other special functions, but that is sort of cheatish I guess. How do you know it's not possible?
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