# How to get closed form solution of transcendental equation? [closed]

Transcendental-Equation (closed form solution)

• I want some method to get closed form solution of a transcendental equation and that is in terms of Lambert's Omega function.
• The equation $$x + e^x = k$$ has a closed form solution $$x = W(k)$$, but what about equation in some other form like $$x + \sin (x) = k$$? Please reply... Thanks in advance.
• It's a good question. – Ordinary users68 Apr 20 at 7:31
• We generally prefer to answer questions where you’ve tried yourself to get a solution with the software Mathematica, but can’t get your code to work....at the current stand this is a question for math.stackexchange.com ...did you intend to ask there? – morbo Apr 20 at 8:00
• In general, one does not expect closed form solutions for transcendental equations. In the case of $x+\exp(x)=k$ and similar equations, the Lambert function (and the related Wright function) had to be created just to represent their solutions, but it is not always applicable. – J. M.'s technical difficulties Apr 20 at 10:22

There are no known closed form analytic solution. However, one can obtain a series solution.

s=InverseSeries[Series[x+Sin[x],{x,0,3}],k]
r[k_]=N[Normal[InverseSeries[Series[x+Sin[x],{x,0,20}],k]]]


It yields

(* k/2+k^3/96+O[k]^5 *)
(* 0.5 k+0.0104167 k^3+0.000520833 k^5+0.0000333271 k^7+2.4005*10^-6 k^9+1.85392*10^-7 k^11+1.49923*10^-8 k^13+1.25265*10^-9 k^15+1.07246*10^-10 k^17+9.35709*10^-12 k^19 *)


A few values are

 Table[r[k],{k,0,3}]


{0.,0.510973,1.10606,2.12739}

I do not see how x+ sin (x) = k this can be solved in closed form. But only for specific value of k and Mathematica can solve it, but gives numerical values.

 Plot[x + Sin[x], {x, -4 Pi, 4 Pi}]


  InputForm@Table[x /. First@Solve[x + Sin[x] == k, x, Reals], {k, 0, 10}]


 N[%]


You could obtain values of k as function of x which satisfies the equation (around $$x=0$$ using

  sol = AsymptoticSolve[x + Sin[x] == k, {k}, {x, 0, 10}]


Around $$x=1$$

  sol = AsymptoticSolve[x + Sin[x] == k, {k}, {x, 1, 10}]


And so on. It does not seem that there is closed form solution for any constant $$k$$. But you could ask in the math group. May be there is some magic someone there can come up with.