# Warning when using NSolve to solve equations

M = 1/10;
sol = NSolve[{n == 1/(2 M) + e/Tan[2 e], e == 1/(2 M)*Sqrt[Exp[4 n] - (1 -
2*M*n)^2], 1/10 < e < 6, -1 < n < 1}, {e, n}];


It gives the warning that " NSolve was unable to prove that the solution set found is complete"

and then plot the functions

ContourPlot[{n == 5 + e/Tan[2 e],
e == 5*Sqrt[Exp[4 n] - (1 - 2/10*n)^2]}, {e, 1/100, 6.3}, {n, -3,
3}, ContourStyle -> {Black, {Dashed, Red}},
Epilog -> {RGBColor[0.2, 0.1, 0.8], Thickness[0.005],
Table[Circle[{e /. sol[[k]], n /. sol[[k]]}, 0.07], {k, 1, 4}]},
Axes -> True, Frame -> False, PlotPoints -> 40,
Prolog -> {Line[Table[{{k*Pi/2, -3}, {k*Pi/2, 3}}, {k, 1, 4}]]}]


it give the picture without circles at the point of intersection.It seems that the code

Epilog -> {RGBColor[0.2, 0.1, 0.8], Thickness[0.005],
Table[Circle[{e /. sol[[k]], n /. sol[[k]]}, 0.07], {k, 1, 4}]}


didn't work

the first is the problematic picture and the second is the targeted picture picture one picture two

• Your code works with version 11.3. What version are you using? May 14 '18 at 16:15

Since e is real, 25 E^(4 n) - (-5 + n)^2 must be greater zero. Than you can solve for 25 E^(4 n) - (-5 + n)^2 == e^2 . Further do

FullSimplify[
ComplexExpand[n == 1/(2 M) + e/Tan[2 e],
TargetFunctions -> {Re, Im}], 1/10 < e < 6 && -1 < n < 1]

(*   -10 + 2 n + e Tan[e] == e Cot[e]   *)


nsol = NSolve[{-10 + 2 n + e Tan[e] == e Cot[e],
25 E^(4 n) - (-5 + n)^2 == e^2, 1/10 < e < 6, -1 < n < 1}, {e,  n},
WorkingPrecision -> 30]

(*   {{e -> 1.43094865810295458384888304493,
n -> 0.0180132370706695599946060476995}, {e ->
2.87757745845758744317117387730,
n -> 0.0665106724899688922422774527454}, {e ->
4.34782164851352692158237679687,
n -> 0.133182763206777152207840057118}, {e ->
5.84137958607605208314996732776,
n -> 0.206480096302156530694105177369}}   *)


Test

{-10 + 2 n + e Tan[e] - e Cot[e],
Sqrt[25 E^(4 n) - (-5 + n)^2] - e} /. nsol

(*   {{0.*10^-28, 0.*10^-30}, {0.*10^-28, 0.*10^-30}, {0.*10^-28,
0.*10^-30}, {0.*10^-28, 0.*10^-29}}   *)

$Version (* "11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018)" *) M = 1/10; sol = NSolve[ {n == 1/(2 M) + e/Tan[2 e], e == 1/(2 M)*Sqrt[Exp[4 n] - (1 - 2*M*n)^2], 1/10 < e < 6, -1 < n < 1}, {e, n}] // Quiet; ContourPlot[ {n == 5 + e/Tan[2 e], e == 5*Sqrt[Exp[4 n] - (1 - 2/10*n)^2]}, {e, 1/100, 6.3}, {n, -3, 3}, ContourStyle -> {Black, {Dashed, Red}}, Epilog -> {RGBColor[0.2, 0.1, 0.8], Thickness[0.005], Circle[{e, n}, 0.07] /. sol}, Axes -> True, Frame -> False, PlotPoints -> 40, Prolog -> {Line[{{#, -3}, {#, 3}} & /@ (Range[1, 4] Pi/2)]}] EDIT: With version 10.3, NSolve is much slower and produces imaginary artifacts that can be removed with Chop $Version

(* "10.3.1 for Mac OS X x86 (64-bit) (December 9, 2015)" *)

M = 1/10;
sol = Cases[
NSolve[
{n == 1/(2 M) + e/Tan[2 e],
e == 1/(2 M)*Sqrt[Exp[4 n] - (1 - 2*M*n)^2],
1/10 < e < 6, -1 < n < 1}, {e, n},
Reals],
_?(FreeQ[#, Undefined] &)] // Chop // Quiet;

ContourPlot[{n == 5 + e/Tan[2 e], e == 5*Sqrt[Exp[4 n] - (1 - 2/10*n)^2]}, {e,
1/100, 6.3}, {n, -3, 3},
ContourStyle -> {Black, {Dashed, Red}},
Epilog -> {RGBColor[0.2, 0.1, 0.8], Thickness[0.005],
Circle[{e, n}, 0.07] /. sol},
Axes -> True,
Frame -> False,
PlotPoints -> 40,
Prolog -> {Line[{{#, -3}, {#, 3}} & /@ (Range[1, 4] Pi/2)]}] • The version I used is 10.3.@Bob Hanlon May 15 '18 at 1:44
• @Number_9527 - edit above added solution using version 10.3.1 May 15 '18 at 2:59