Hi:
Is there an algebraic means by which to solve this equation and others like it? I can estimate via Newton's method but I would like to know how to determine solution(s) exactly.
e^x + 2x = 7
Thank you.
Rich B.
Hi Ron:
I followed you right up to the last line. I don't know what W(...) means. Is that 'function W of the argument 0.5e^3.5'? And if so, what is the function W? Confused...
I will be away for a week, effective Saturday, A.M., so if I don't get right back with the proper 'thanks for filling in the gaps', you will understand I trust.
P.Hckr: Thank you for your response. I gave your work a quick read upon getting in at 1:30 A.M., and I will definitely need a good night's sleep before re-reading and digesting.
Thank you both,
Rich B.
W is Lambert's W function. W(z) is the inverse function of:
f(w)=w exp(w)
So if z=w exp(w), then w=W(z). It is considered by a number of people that
it ought to be included in the list of elementary transcendental functions.
You can learn more about it here.
That W cannot be written as a finite combination of the other elementary
functions and algebraic operations (as far as I know anyway) shows that
a general equation of the form:
exp(x) + ax + b = 0
cannot be solved using just the normal elemantary function and algebraic
operations in a finite closed form.
RonL