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.
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Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.
Abstract
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.
Keywords:
Power Exponential Function, Equivalent Function, Approximation
1. Introduction
The literature devoted to the equation , , is really limited. From[1] 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[2] writes a discussion of the curve defined by , , and recently Y. S. Kupitz and H. Martini[3] demonstrate the following two propositions: (1) There is a nontrivial solution to the equation , , if and only if , and for such a the solution is unique, and (2) The only nontrivial 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  (1) 
admits the trivial solution, which will be named as solution (A),  (2) 
and another solution (B), which may be found either by successive iterations or by using some software, like Mathematica[4], in a computer.Obviously, solution (B) is symmetrical with respect to the straight line defined by solution (A).
2. NonTrivial Solution (B)
To find out the solution (B) one can proceed as follows:From (1)  (3) 
 (4) 
 (5) 
 (6) 
 (7) 
And,  (8) 
being ProductLog[z] the function which gives the principal solution for w in  (9) 
as defined and tabulated by Mathematica.Then solution (B) may be tabulated from  (10) 
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 Y_{B} looks very close to the hyperbola  (11) 
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 Y_{B}, a third column (Y_{H1}) is added in Table I, showing  (12) 
as given by (11), whereas the fifth column shows the distance Y_{H1} – Y_{B}.Then, accounting for the fact that the curve also goes through the point (e, e), the hyperbola  (13) 
is considered too and  (14) 
as given by (13), is shown in the fourth column of table 1, whereas the distance Y_{H2}  Y_{B} appears in the sixth column.  X  Y_{B}  Y_{H1}  Y_{H2}  Y_{H1}Y_{B}  Y_{H2}Y_{B}  e  e  2.7459  2.7183  0.0276  0.0000  2.8  2.6405  2.6667  2.6403  0.0262  0.0002  2.9  2.5548  2.5790  2.5539  0.0242  0.0009  3.0  2.4781  2.5000  2.4763  0.0219  0.0018  3.5  2.1897  2.2000  2.1810  0.0103  0.0087  4.0  2.0000  2.0000  1.9842  0.0000  0.0158  4.5  1.8655  1.8571  1.8436  0.0084  0.0219  5.0  1.7649  1.7500  1.7381  0.0149  0.0268  6.0  1.6242  1.6000  1.5905  0.0242  0.0337  7.0  1.5301  1.5000  1.4921  0.0301  0.0380  8.0  1.4625  1.4286  1.4218  0.0339  0.0407  9.0  1.4114  1.3750  1.3691  0.0364  0.0423  10.0  1.3713  1.3333  1.3281  0.0380  0.0432  12.0  1.3122  1.2727  1.2684  0.0395  0.0438  14.0  1.2707  1.2308  1.2271  0.0399  0.0436  16.0  1.2396  1.2000  1.1968  0.0396  0.0428  18.0  1.2155  1.1765  1.1737  0.0390  0.0418  20.0  1.1962  1.1579  1.1554  0.0383  0.0408  25.0  1.1613  1.1250  1.1230  0.0363  0.0383  30.0  1.1377  1.1034  1.1018  0.0343  00359  35.0  1.1206  1.0882  1.0868  0.0324  0.0338  40.0  1.1075  1.0769  1.0757  0.0306  0.0318  45.0  1.0973  1.0682  1.0671  0.0291  0.0302  50.0  1.0889  1.0612  1.0603  0.0277  0.0286  60.0  1.0762  1.0508  1.0500  0.0254  0.0262  70.0  1.0669  1.0435  1.0428  0.0234  0.0241  80.0  1.0598  1.0380  1.0374  0.0218  0.0224  90.0  1.0541  1.0337  1.0332  0.0204  0.0209  100.0  1.0495  1.0303  1.0298  0.0192  0.0197  125.0  1.0410  1.0242  1.0238  0.0168  0.0172  150.0  1.0352  1.0201  1.0198  0.0151  0.0154  175.0  1.0309  1.0172  1.0170  0.0137  0.0139  200.0  1.0276  1.0151  1.0148  0.0125  0.0128  250.0  1.0228  1.0120  1.0119  0.0108  0.0109  300.0  1.0196  1.0100  1.0099  0.0096  0.0097  400.0  1.0153  1.0075  1.0074  0.0078  0.0079  500.0  1.0127  1.0060  1.0059  0.0067  0.0068 


Thus, direct reading of table I shows that the hyperbola (11) is closer to Y_{B} than the hyperbola (13), and that  (15) 
for two reasons: 1) this value is not reached before , and 2) for and onwards the distance between Y_{B} and the asymptote , as well as between Y_{H1}_{ }and the same asymptote, is less than 0.04, which implies (15).In fact, in figure 1 the points representing Y_{H1} are plotted over the curve Y_{B}_{ }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 
4. Conclusions
The little difference between the two functions Y_{H1} and Y_{B}, which remains always under 0.04, means that the much simpler hyperbola given by equation (11) is a very good approximation to the implicit powerexponential function defined by equation (1).
References
[1]  R. C. Archivald, “Problem notes, No. , Amer. Math. Monthly, vol. 28, pp. 141143, 1921 
[2]  E. J. Moulton, “The real function defined by x ^{y} = y^{x} “, Amer. Math. Monthly, vol. 23, pp. 233237, 1916 
[3]  Y. S. Kupitz, and H. Martini, C. “On the equation x^{y }= y^{x }”, Elemente der Mathematik, vol. 55, pp. 95–101, 2000 
[4]  Mathematica, Trade Mark. Wolfram Research, Inc., 
[5]  Mitteldorf, “Solutions to x ^{y} = y^{x} “ [on line]. Available from: http://mathforum.org/library/drmath/view/53229.html (accessed November 2011) 
[6]  Vogler, “Solving the equation x ^{y} = y^{x} “[on line]. Available from: http://mathforum.org/library/drmath/view/66166.html (accessed November 2011) 