# Solution of a system of three equations, one equation involving the real part of an integral

Let $$a,b,c$$ and $$t$$ be complex numbers. How can one solve the following three equations for $$a,b,$$ and $$c$$ in terms of $$t$$:

$$a^2 + b^2 + c^2 + 2 t =0,$$ $$a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 b^2 c^2 = 16,$$ and $$\Re\left(\int^b_a \sqrt{(z^2-a^2)(z^2-b^2)(z^2-c^2)} \ \mathrm{d}z \right)=0.$$ My attepmts using Solve, and NSolve have failed (even for a fixed $$\tau$$). Any help is very much appreciated.

My Attempts

K[a_, b_, c_] := Re[Integrate[Sqrt[(s^2 - a^2) (s^2 - b^2) (s^2 - c^2)], {s, a, b}]]

Solve[{a^2 + b^2 + c^2 + 2 t == 0, a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 c^2 b^2 == 16, K[a, b, c] == 0}, {a, b, c}]


But it does not yield any ansswer. I get the error

Solve::nsmet: This system cannot be solved with the methods available to Solve.


These equations should have a solution, in particular, for any $$t=\mathrm{i} y$$, with $$y>\sqrt{12}$$. So I tried it with $$y=4$$

NSolve[{a^2 + b^2 + c^2 + 2 (4 I) == 0,
a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 c^2 b^2 == 16,
K[a, b, c] == 0}, {a, b, c}]


But I get the error

NSolve::nsmet: This system cannot be solved with the methods available to NSolve.

• It would be more helpful if you a) showed your euqations as MMA code, not Tex, so we don't have to type them in again; b) showed the exact code you tried so far. Dec 17 '20 at 19:06
• @MarcoB Thanks for your comment. I edited the post. Dec 17 '20 at 19:36

First, we specify t. Second, we reduce the system under consideration to the reals by the change a=a1+I*a2,b=b1+I*b2,c=c1+I*c2 and extracting the real and imaginary parts. Third, in order to solve that system in 6 variables, we find the minimum of the sum of the squared LHSes.

t = -5; NMinimize[Re[NIntegrate[
Sqrt[(z^2 - (a1 + I*a2)^2)*(z^2 - (b1 + I*b2)^2)*(z^2 - (c1 +
I*c2)^2)], {z, a1 + I*a2, b1 + I*b2}]]^2 +
ComplexExpand[Re[(a1 + I*b2)^2 + (b1 + I*b2)^2 + (c1 + I*c2)^2 + 2*t]]^2 +
ComplexExpand[Im[(a1 + I*b2)^2 + (b1 + I*b2)^2 + (c1 + I*c2)^2 +
2*t]]^2 + (ComplexExpand[Re[(a1 + I*b2)^4 + (b1 + I*b2)^4 + (c1 + I*c2)^4 -
2*(a1 + I*b2)^2*(b1 + I*b2)^2 - 2*(a1 + I*b2)^2*(c1 + I*c2)^2 -
2*(b1 + I*b2)^2*(c1 + I*c2)^2]] - 16)^2 +
ComplexExpand[Im[(a1 + I*b2)^4 + (b1 + I*b2)^4 + (c1 + I*c2)^4 -
2*(a1 + I*b2)^2*(b1 + I*b2)^2 - 2*(a1 + I*b2)^2*(c1 + I*c2)^2 -
2*(b1 + I*b2)^2*(c1 + I*c2)^2]]^2, {a1, b1, c1, a2, b2, c2},
Method -> "DifferentialEvolution"]


{1.17036*10^-15, {a1 -> 2.64435, b1 -> 1.73702, c1 -> 2.79462*10^-9, a2 -> 5.46634*10^-10, b2 -> 7.91073*10^-10, c2 -> 0.0991094}}

• For t=4*I the usage of the options Method -> {"DifferentialEvolution", "ScalingFactor" -> 1}, AccuracyGoal -> 4, PrecisionGoal -> 5] gives {4.54936*10^-14,{a1->1.388,b1->1.17528,c1->-0.839586,a2->1.00926,b2->-1.30061,c2->0.793434}}. Dec 17 '20 at 21:09