Re: A *very* didactic little quiz Message #40 Posted by Valentin Albillo on 19 July 2004, 5:41 a.m., in response to message #39 by GE (France)
Hi, GE:
GE posted:
"Nice puzzle, Valentin (and lots of replies)"
Yes, thanks for your kind words, I'm very happy people were insterested, as I feel it makes a pretty fine example of didactic-yet-entertaining mathematics, if I may coin a term.
"There is something to take into account : risk. "
Pleeeease, GE ! :-) It's only a pure-math quiz, I said so much in my original post. No inflation, risk, physical feasibility, speed-of-light unattainable for non-zero mass objects, not so much gold in the Universe, the Earth would be crushed, gold would be worth nothing, the bank teller would run out with the money, etc, etc, etc, etc, etc, etc, etc, etc ! :-) We do have a very descriptive Spanish term for this, but it's very graphic and I don't know how to properly translate it. I'm sure you do have some equivalent in French ;-)
"In Maple, solving the second form brings an expression using the funtion "LambertW()". Does anyone know what this is ? Pointers welcome. "
The Lambert-W function (known in Mathematica as "ProductLog") is an extremely interesting function, that appears everywhere and it's been unofficially nominated for the next "standard elementary function", similar to trigonometric and exponential functions, so it's not that impossible that we'll have calculators including it as a standard function in the near future. The advantage to consider it a standard function is that you can then express a large number of roots to trascendental equations, integrals, and ODEs, to name a few, in closed form in terms of LambertW (instead of having non-closed, infinite series expansions or sums), which is exactly what we do achieve by using sin, cos, exp, log, etc.. I've certainly used it a lot, to express roots of equations in terms of it (for instance, the solution to x^x = a can be expressed in terms of LambertW),
like this:
x^x = a -> x = e^W(ln(a))
e.g.: x^x = Pi -> x = e^W(ln(Pi) = 1.85410596792
HP-71B code: first define
10 DEF FNW(X)=FNROOT(-1,10,FVAR*EXP(FVAR)-X)
then:
> X=EXP(FNW(LN(PI)))
> X
1.85410596792
> X^X
3.14159265358
As for your equation, e^x/x = C, the solution is:
e^x/x = C -> x = -W(-1/C)
HP-71B code, assuming the previous definition of FNW(X):
e.g: e^x/x = 5 -> x = -W(-1/5)
>X=-FNF(-1/5)
>X
0.259171101818
>EXP(X)/X
5
The best, most comprehensive and interesting document I know about LambertW is available from here, in PDF format. Just go down the page to the Some interesting math-related papers section and click the first link, "On the Lambert-W function".
Best regards from V.
Edited: 19 July 2004, 8:36 a.m.
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