1
$\begingroup$

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.
$\endgroup$
  • $\begingroup$ It's a good question. $\endgroup$ – Ordinary users68 Apr 20 at 7:31
  • $\begingroup$ 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? $\endgroup$ – morbo Apr 20 at 8:00
  • 5
    $\begingroup$ 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. $\endgroup$ – J. M.'s technical difficulties Apr 20 at 10:22
4
$\begingroup$

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}

| improve this answer | |
$\endgroup$
3
$\begingroup$

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}]

Mathematica graphics

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

Mathematica graphics

 N[%]

Mathematica graphics

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}]

Mathematica graphics

Around $x=1$

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

Mathematica graphics

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.

| improve this answer | |
$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.