How can I find a root of my transcendental equation near a certain point?

Question

What function could I use to solve this equation for x:

E^(-x^2) = 1 - Cos[x]

I tried Solve, Reduce, Root... None of them worked.

FindInstance gave me an imaginary answer.

Adding Reals, gave me:

{{x -> Root[{-1 + E^-#1^2 + Cos[#1] &, -6.28318531096301494721675292506}]}}

but this is not the answer I wanted, which would be between 0 and 1.

How would I solve this?

E.g. Solve[E^(-x^2) == 1 - Cos[x] && 0 <= x <= 1, x] yields {{x -> Root[{1 - E^#1^2 + E^#1^2 Cos[#1] &, 0.94194408148019155746}]}}. See e.g. How do I work with Root objects? — Artes, Apr 22 '14 at 22:14
The reason Solve has a hard time is that it tries to find all solutions, but there are actually infinitely many of them: two each surrounding $x=2n\pi$ for all $n\in\mathbb Z$. (So WolframAlpha's answer is misleading.) Restricting the domain, as in Artes' comment, helps because then there are only a finite number of solutions. — , Apr 22 '14 at 22:17
Take a look also at How do I solve this equation? and Can Reduce really not solve for x here?. — Artes, Apr 22 '14 at 22:24

score 3 · Answer 1 · answered Apr 23 '14 at 00:20

3

This?

NSolve[E^(-x^2) == 1 - Cos[x] && 0 < x < 1, x]
(* {{x -> 0.941944}} *)

Or this?

FindRoot[E^(-x^2) - (1 - Cos[x]), {x, 1}]
(* {x -> 0.941944} *)

answered Apr 23 '14 at 00:20

Michael E2

1 Answers1