Why the Lambert W function is more famous than the inverse of x^x?

[u/i-dunno-who-i-am]

The inverse of xe^x is the Lambert W function, whereas the inverse of x^x is called ssrt(x) in wiki I think. They can also be expressed in terms of each other.

I am wondering why the Lambert W function is more popular than the inverse of x^x, when tetration is the 4th hyperoperations(+,x, exponentiation, then tetration). Since we also study the inverses of addition, multiplication, exponentiation, but why not the inverse of tetration?

[u/DenPanserbjorn]

xe^x seems more applicable especially with solutions of common ODE’s.

Also we can rewrite ssrt in terms of W: x^x = e^(xlnx) = e^(lnx * e^lnx)
So ssrt(x) = e^W(ln(y))

Also, tetration isn’t really an operation most people learn because it really doesn’t have many practical applications.

[u/i-dunno-who-i-am → u/DenPanserbjorn]

Can you provide example of ODEs that uses xe^x? I haven't studied ODE until that level so idk much about that kind of ODE

[u/defectivetoaster1 → u/i-dunno-who-i-am]

the ODE y’’ -2y’ + y = 0 has the general solution y=Ae^x + Bxe^x . In general any linear ODE with constant coefficients whose characteristic reaction has repeated roots will see terms in the general solution that are exponentials (possibly complex) multiplied by a power of x

[u/Farkle_Griffen2 → u/i-dunno-who-i-am]

You should not be getting downvoted for asking math questions on a sub literally called r/askmath.

[u/ExcelsiorStatistics → u/i-dunno-who-i-am]

One simple example that comes up in real-world applications: if the chance of an event occurring increases linearly with time (e.g., stress accumulating continuously on a fault before an earthquake), the PDF of the interval between events is te^-t shaped (a "Rayleigh distribution," a special case of the Weibull distribution. In general if the hazard is a polynomial, the resulting pdf is Weibull.)

This amounts to solving F'(t) / (1-F(t)) = t subject to F(0)=0.

[u/dancingbanana123]

Realistically, I think most people have just been introduced to Lambert W functions through wolfram alpha. I've never had a class talk about them. It's just one of those neat things you pick up on your own I guess. I've never really seen any application of either of them other than just to emphasize how gross exponential equations can be to solve.

[u/pruvisto → u/dancingbanana123]

I ran into it once in the context of worst-case behaviour of height-balanced trees. With a height difference bound of d, the minimum number of elements in a tree of height h is roughly B_d * C_d ^ h, and the growth of the constants with varying d is C_d ~ 1 + W(d)/d and B_d → d / (W(d) * (1 + W(d))).

Yeah I know, this is pretty niche stuff.

[u/DrBadRudes → u/dancingbanana123]

It's used in the solution of the 1d schroedinger equation with 2 delta wells.

[u/patenteng → u/dancingbanana123]

The Lambert W function can be used to solve the voltage of a diode in series with a resistor. The current through the diode is an exponential function with respect to the voltage across the diode. So you get something like v + e^v = E, where E is the supplied voltage.

[u/dancingbanana123 → u/patenteng]

Yeah I guess I mean outside the context of applied math, since approximation methods become incredibly useful there (not to say applied math doesn't count ofc)

[u/haydencoffing]

I think W is more common than the super square root due to two things: one is the preference of many mathematicians to have exponents in base e. Two is the W function is not just one function, but a family of functions as there are infinitely many functions of the same property.