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 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

(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. Non-Trivial 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.

Table 1. X YB YH1 YH2 YH1-YB YH2-YB 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 -0-0359 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 power-exponential function defined by equation (1).