Finding the real solutions using the ones from a complexification

Question

Let

$$f(x) = 2(1+\cos x - x \sin x) \, .$$

The equation $f(x)=0$ has a set of solutions given by $x = (2n - 1)\pi, \, n \in \mathbb{Z}$, easy to find since $\sin x = 0 $ at these roots. To find all solutions we need to solve the equation without any immediate tricks. Reduce and Solve are unable to deal with algebraic solutions, so

NSolve[2 (1 + Cos[x] - x Sin[x]) == 0 && 0 < x < 10, x] => {{x -> 1.30654}, {x -> 3.14159}, {x -> 6.58462}, {x -> 9.42478}}

Now, it'd be the happiest day of my life to find a complete algebraic solution. I decided to see what I could extract from complexifying $f$, that is,

$$\tilde{f}(\tilde{x}) = 2(1+ e^{i \tilde{x}} + i \tilde{x} e^{i \tilde{x}} ) \, ,$$

where the $\tilde{}$ denotes a complex number (notice both $f$ and $x$ are complexified). It is evident both that $\mathrm{Re}[\tilde{f}] = f$ and that $\tilde{f} = \mathrm{Re}[\tilde{f}] + i\, \mathrm{Im}[\tilde{f}]$.Reduce works for this case:

Reduce[2 (1 + Exp[I x] + I x Exp[I x]) == 0, x] => C[1] \[Element] Integers && x == I - I ProductLog[C[1], -E]

but I have no idea whether or not it is possible to find solutions for the real equation from this complex solution set or if it's just plain useless. Can someone help?

P.S.: I had problems to decide whether or not this question was suited for Math.SE (since it is probably about very basic complex analysis), but since I'm using Mathematica to find the solutions I decided to post it here. Feel free to move it in case it doesn't belong here.

Answer 1

You can't get a complete algebraic rule for the solution, since there is no one for x == Cot[x/2]

Factor the f[x] into 3 factors with TrigFactor

g[x_] = f[x] // TrigFactor

(*     -4 Cos[x/2] (-Cos[x/2] + x Sin[x/2])     *)

Solve the first part -4 Cos[x/2] for x to get the known general rule x = (2 n-1) Pi

Solve[g[x][[1 ;; 2]] == 0, x, Reals]

(*     {{x -> ConditionalExpression[2 (-(\[Pi]/2) + 2 \[Pi] C[1]), 
C[1] \[Element] Integers]}, {x -> 
              ConditionalExpression[2 (\[Pi]/2 + 2 \[Pi] C[1]), 
C[1] \[Element] Integers]}}     *)

The third factor is x == Cot[x/2] and roots do not follow a general rule, but can be solved for a restricted x-range

Solve[g[x][[3]] == 0 && 0 <= x < 30, x, Reals]

(*     {{x -> Root[{Cos[#1/2] - Sin[#1/2] #1 &, 
 1.30654237418880620223}]}, {x -> 
   Root[{Cos[#1/2] - Sin[#1/2] #1 &, 6.5846200425641731922}]}, {x -> 
   Root[{Cos[#1/2] - Sin[#1/2] #1 &, 12.7232407841313299947}]}, {x -> 
   Root[{Cos[#1/2] - Sin[#1/2] #1 &, 18.9549714108415918088}]}, {x -> 
   Root[{Cos[#1/2] - Sin[#1/2] #1 &, 25.212026888550825656}]}}     *)

Answer 2

eqn = 2 (1 + Cos[x] - x Sin[x]) == 0;

Selecting the solutions within the interval that are not Root objects

soln1 = Select[x /. Solve[{eqn, 0 <= x <= 50}, x], FreeQ[#, Root] &]

(* {π, 3 π, 5 π, 7 π, 9 π, 11 π, 13 π, 15 π} *)

Using FindSequenceFunction to generalize the sequence

soln2 = FindSequenceFunction[soln1, n] // Simplify

(* (-1 + 2 n) π *)

Verifying the solution for all integers

Assuming[Element[n, Integers], eqn /. x -> soln2 // Simplify]

(* True *)

This does not include the solutions represented as Root objects.