# Question:Solving a system of equations

## Question:Solving a system of equations

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Solving a system of equations

I want to solve the following system (please see the code) for small even N=4,8,16. They said that the roots lie on the lines Im x_i = ±π/2 and they are symmetric with respect to the imaginary axis. For N=4, I tried with fsolve directly but the roots are:

x[1] = -3.799588601-1.570796327*I,

x[2] = -3.799588601-1.570796327*I,

x[3] = .1196098914-1.570796327*I,

x[4] = 1.110549099-1.570796327*I

these solutions are not valid because half of them should have +π/2 as an imaginary part, and they should be mutually distinct and also symmetric with respect to the imaginary axis. Therefore, for N=4  I am expecting solutions of these form :

x[1] = a_1-I* π/2,

x[2] = -a_1-I* π/2,

x[3] = a_1+I* π/2,

x[4] = -a_1+I* π/2,

I tried with fslove({system}, {Im(x[1]) = .5 .. 1, Im(x[2]) = .5 .. 1, Im(x[3]) = -1 .. -.5, Im(x[4]) = -1 .. -.5}) but it does not work! I do not know what should I do to make Fsolve locate the wanted solutions. Thanks in advance!

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