We need a similar function to solve Kepler's equation! *M* = *E* - e sin *E .* http://www.tweedledum.com/rwg/pizza.htm Will this suffice to describe motion in an elliptical orbit? E.g., {A cos(u(t)), B sin(u(t))}. What is u(t)? Is it the same, modulo scaling t, for all orbits of the shape "A by B"? Does scaling the "sun"'s gravity just scale t somehow? —rwg Date: 2019-08-12 13:11 From: Andres Valloud <avalloud@smalltalk.comcastbiz.net> To: math-fun <math-fun@mailman.xmission.com> Reply-To: "andres.valloud@gmail.com" <andres.valloud@gmail.com>, math-fun < math-fun@mailman.xmission.com> Thanks! The Wikipedia page notes W cannot be expressed in terms of elementary functions. No wonder trying to clear out x from the original equation was rather difficult! On 8/12/19 2:37 , Neil Bickford wrote:

I think Example 1 in https://en.wikipedia.org/wiki/Lambert_W_function#Example_1 might be applicable to this!

For instance, for a x + b = ln x, set y = ln x - b to get (a e^b) e^y = y. Then -a e^b = -y e^(-y), and so y = -W(-a e^b). Substituting back then gives x = e^(b-W(-a e^b)), assuming I’ve done my math right!

Mathematica informs me that since a x + b = ln x, this can also be written as x = -W(-a e^b)/a.

--Neil Bickford

On Mon, Aug 12, 2019 at 2:11 AM Andres Valloud <avalloud@smalltalk.comcastbiz.net <mailto:avalloud@smalltalk.comcastbiz.net>> wrote:

What's a neat way to solve equations such as the below exactly?

ax + b = ln x

In particular, what to do when b = 0?

Andres.