Solving an implicit equation involving Elliptic integral

Question

I have the following equation:

EllipticE[1/(1 + 16*(ja*me + jb*mo)^2)] + 
  16*(ja*me + jb*mo)^2*EllipticK[1/(1 + 16*(ja*me + jb*mo)^2)] == 0

I want to find values of me and mo. I have tried to find these by using Solve and FindRoot but I didn't get anything. Is there any other way to solve it? And how do I plot mo or me as a function of ja and jb?

Answer 1

Clear["Global`*"]

eq1 = EllipticE[1/(1 + 16*(ja*me + jb*mo)^2)] + 
    16*(ja*me + jb*mo)^2*EllipticK[1/(1 + 16*(ja*me + jb*mo)^2)] == 0;

As recommended by @Bill,

eq2 = eq1 /. ja*me + jb*mo :> u

(* EllipticE[1/(1 + 16 u^2)] + 16 u^2 EllipticK[1/(1 + 16 u^2)] == 0 *)

FindInstance[eq2, u, Reals]

(* FindInstance::nsmet: The methods available to FindInstance are insufficient 
  to find the requested instances or prove they do not exist. *)

Removing the domain constraint,

(sol = FindInstance[eq2, u, 2]) // N

(* {{u -> 0. - 0.322665 I}, {u -> 0. + 0.322665 I}} *)

Verifying the solution,

eq2 /. sol // FullSimplify

(* {True, True} *)

Graphically,

ComplexPlot3D[eq2[[1]], {u, -1 - I, 1 + I},
 AxesLabel -> {Re, Im, None}]

eq3 = eq2 /. u -> I*y

(* EllipticE[1/(1 - 16 y^2)] - 16 y^2 EllipticK[1/(1 - 16 y^2)] == 0 *)

FindRoot[eq3, {y, #}] & /@ {-0.3, 0.3}

(* {{y -> -0.322665}, {y -> 0.322665}} *)

Graphically,

Plot[Evaluate@eq3[[1]], {y, -1, 1}]

Answer 2

The equation can be reduced to a (complex) quadratic equation in four variables. One may obtain one variable in terms of the other three, but one may not obtain two of the variables in terms of the other two, as requested by the OP.

eqn = EllipticE[1/(1 + 16*(ja*me + jb*mo)^2)] + 
    16*(ja*me + jb*mo)^2*EllipticK[1/(1 + 16*(ja*me + jb*mo)^2)] == 0;

(* reduction to a quadratic equation *)
ueqn = eqn /. First@Solve[u == 16*(ja*me + jb*mo)^2, ja] // Simplify

(*  EllipticE[1/(1 + u)] + u EllipticK[1/(1 + u)] == 0  *)

usol = Solve[ueqn && -2 < u < -1, u]
eqn2 = Replace[
  usol, {{r_Rule}} :> (Equal @@ r /. 
     First@Solve[u == 16*(ja*me + jb*mo)^2, u])]

(* 
{{u -> Root[{EllipticE[1/(1 + #1)] + 
       EllipticK[1/(1 + #1)] #1 &, -1.665807876436924963209380412862432}]}}

(* quadratic equation *)
16 (ja me + jb mo)^2 == 
 Root[{EllipticE[1/(1 + #1)] + 
     EllipticK[1/(1 + #1)] #1 &, -1.665807876436924963209380412862432}]
*)

Note the exact Root[] object is negative, which means the solutions are complex. The solution space is a three-dimensional complex manifold (six real dimensions) in four-dimensional complex space (eight real dimensions). I don't know how one would graphically represent such a solution space, but algebraically, it is not hard to deal with.

Numerical/graphical evidence of only one solution

One solution to ueqn inside Abs[u] <= 2.5:

Show[
 ComplexPlot3D[EllipticE[1/(1 + u)] + u EllipticK[1/(1 + u)],
  {u, -3 - 3 I, 3 + 3 I},
  ColorFunction -> {ColorData["Rainbow"][Clip[#7/4]] &, None}
  , ColorFunctionScaling -> False,
  MeshFunctions -> {Re[#1] & , Im[#1] & , Abs[#1] &}
  , Mesh -> {{0, -1, -2}, {0}, {2.5}}
  , MeshStyle -> {Automatic, Automatic, Thick},
  PlotRange -> {0, 10}],
 ComplexPlot3D[0, {u, -3 - 3 I, 3 + 3 I},
  ColorFunction -> {RGBColor[0.5, 0.75, 1., 0.5] &, None}, 
  ColorFunctionScaling -> False,
  MeshFunctions -> {Re[#1] & , Im[#2] & }, PlotRange -> {0, 10}]
 ]

No solutions outside `Abs[u] 5/u`):

ComplexPlot3D[EllipticE[1/(1 + 5/u)] + 5/u EllipticK[1/(1 + 5/u)],
 {u, -3 - 3 I, 3 + 3 I},
 ColorFunction -> {ColorData["Rainbow"][Clip[#7/4]] &, None}
 , ColorFunctionScaling -> False,
 MeshFunctions -> {Re[5/#1] & , Im[5/#1] & , Abs[5/#1] &}
 , Mesh -> {{0, -1, -2}, {0}, {2.5}}
 , MeshStyle -> {Automatic, Automatic, Thick}]

It is probably not hard to prove that there are no other solutions, but I'll leave that to others. For instance, EllipticE[1/(1 + u)] and EllipticK[1/(1 + u)] are bounded for u sufficiently large, of course. In fact, EllipticE[1/(1 + 5/u)] + 5/u EllipticK[1/(1 + 5/u) is asymptotic to (Pi/2) u, so there are no zeros for large u.