José María Mínguez
Dpto. de Física Aplicada II, Universidad de Bilbao, Bilbao, 48930, Spain
Correspondence to: José María Mínguez , Dpto. de Física Aplicada II, Universidad de Bilbao, Bilbao, 48930, Spain.
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
This short paper deals with the implicit function , X,Y > 0, and shows surprinsingly how accurately it is equivalent to another very much simpler and explicit function.
Power Exponential Function, Equivalent Function, Approximation
The literature devoted to the equation , , is really limited. From we know that L. Euler treated it and gave a parametric representation, from which the rational solutions were drawn. He also deduced the existence of the two asymptotes ( and ) to the curve. The same paper gives notice that also Daniel Bernouilli found the rational solutions. Later E. J. Moulton writes a discussion of the curve defined by , , and recently Y. S. Kupitz and H. Martini demonstrate the following two propositions: (1) There is a non-trivial solution to the equation , , if and only if , and for such a the solution is unique, and (2) The only non-trivial integer solutions to the equation , , are (2, 4) and (4, 2).Recently this function has also focussed the attention of mathematicians[5,6], although little has been added to its knowledge and development.In brief, it is well known that the implicit power- exponential function
admits the trivial solution, which will be named as solution (A),
and another solution (B), which may be found either by successive iterations or by using some software, like Mathematica, in a computer.Obviously, solution (B) is symmetrical with respect to the straight line defined by solution (A).
2. Non-Trivial Solution (B)
To find out the solution (B) one can proceed as follows:From (1)
being ProductLog[z] the function which gives the principal solution for w in
as defined and tabulated by Mathematica.Then solution (B) may be tabulated from
Both, equation (10) and direct iterations, yield the results shown in Table I, by means of which figure 1 represents the solution (B) (continuous line), together with solution (A) (discontinuous line).
3. Equivalent Function
Figure 1 shows at first glance that the function YB looks very close to the hyperbola
which, by the way, also admits the integer solutions (2, 4) and (4, 2) as equation (1).In order to analyse how close the function (11) is to the original function YB, a third column (YH1) is added in Table I, showing
as given by (11), whereas the fifth column shows the distance YH1 – YB.Then, accounting for the fact that the curve also goes through the point (e, e), the hyperbola
is considered too and
as given by (13), is shown in the fourth column of table 1, whereas the distance YH2 - YB appears in the sixth column.
Thus, direct reading of table I shows that the hyperbola (11) is closer to YB than the hyperbola (13), and that
for two reasons: 1) this value is not reached before , and 2) for and onwards the distance between YB and the asymptote , as well as between YH1 and the same asymptote, is less than 0.04, which implies (15).In fact, in figure 1 the points representing YH1 are plotted over the curve YB and the closeness is very evident.
|Figure 1. Trivial solution (A) (discontinuous straight line) and solution (B) (full line curve). Overlapping the curve the dots representing the equivalent hyperbolic function|
The little difference between the two functions YH1 and YB, which remains always under 0.04, means that the much simpler hyperbola given by equation (11) is a very good approximation to the implicit power-exponential function defined by equation (1).
| ||R. C. Archivald, “Problem notes, No. , Amer. Math. Monthly, vol. 28, pp. 141-143, 1921|
| ||E. J. Moulton, “The real function defined by x y = yx “, Amer. Math. Monthly, vol. 23, pp. 233-237, 1916|
| ||Y. S. Kupitz, and H. Martini, C. “On the equation xy = yx ”, Elemente der Mathematik, vol. 55, pp. 95–101, 2000|
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