Hi, GE:
GE posted:
" This equation can be related to the definition of the Lambert 'W' function (see V. Albillo's excellent site for more) [...] If the very useful LambertW function were present on the keyboard of our beloved calcs, this kind of problem would be straightforward. I have always believed
that LambertW belongs there as well as Gamma, or even Sin and log."
Thanks for your kind comments. I do agree about Lambert's W, I think it actually belongs with the rest of usual transcendental functions. Of course, such well known functions as sin, cos, exp, and gamma do have thousands of applications where they're useful, but then they've been well known and studied for many centuries, while W's been under the limelight for only a handful of years.
Yet, more and more applications of W are being discovered continually, in a wide variety of engineering and mathematical fields, which only heightens its importance and further raises the general appreciation of its 'elementariness'.
As for it being included as a standard calculator function, I don't think this is likely to happen. Consider, for example, Bessel functions or elliptic functions. They do have tons of important applications, yet I've never seen a calculator with a built-in, keyboard elliptic sine. You get the regular 'circular' sine, and even the hyperbolic sine. But not the elliptic variety. So, Lambert's W isn't likely to make it.
That said, some HP models allow for user-defined functions that are as close as built-in in their handling as to be transparent to the user. For the HP-71B, for instance, you can enter:
DEF FNW(X) = FNROOT(0,10,FVAR*EXP(FVAR)-X)
as a program line. Then, from the keyboard, you can use W(x) as easily as sin(x). Say,to solve:
x^x = 5
you simply would key in:
EXP(FNW(LN(5))) [ENTER]
which would promptly return to the display the value:
2.12937248276
which solves the equation. Notice that you've simply used FNW exactly as you use EXP or LN, i.e., as a built-in function, transparently to the user.
This is as near as it comes to have W directly in the keyboard.
Best regards from V.