Yes Sir, I have a complex binomial series, when expanded. Let me try to implement your advice.

It seems the Binomial series that I calculated manually can't get generated by the inbuilt Binomial function. when I calculated manually I substituted

Omega = Omega_R + i Omega_i

I got

Omega^2=Omega_R^2[1+4IOmega_i - 2i Omega_i Omega_R-3Omega_i^2+2Omega_i^2 Omega_R]

I'm unable to generate this series in Mathematica. not sure which function can generate this series.

Sincerely,

One Eulerian identity of many is that if x is any real number (and under usual other restrictions on the remaining)

( Cos[n x] + I Sin[n x] )^n == ( Cos[n x] + I Sin[n x]) == E^(I n x)

Using the above formula (with crafty substitutions) could lead to brief solution in E^(i n x), compared to the binomial theorem.

Offering spurious theory. In general if a complex function f[z] is defined in Omega it is analytic (holomorphic) in Omega. All real polynomial are analytic. Such an f[z] has a cauchy formula in Omega; roughly equivalent to the taylor polynomial.

Solve[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.

Solve[expr&&vars[Element]Reals,vars,Complexes] solves for real values of variables, but function values are allowed to be complex.

Solve[(z^7 - 1) (2 z^7 - 4) == 0, z, Complexes]

As to your Binomial theorem I still don't know which theorem(s) you are trying to utilize, which i had asked you identify. However I feel that there are theorems covering 7th order complex polynomial built-in to Mathematica, which was displayed in various ways.

In[177]:= 
Re[z]^7 (1 + 14 I Im[z] - 63 Im[z]^2 - 7 I Im[z] Re[z] + 
    42 Im[z]^2 Re[z]) /. z -> (1 + I)

Out[177]= -20 + 7 I

ComplexPlot[(Re[
     z]^7 (1 + 14 I Im[z] - 63 Im[z]^2 - 7 I Im[z] Re[z] + 
      42 Im[z]^2 Re[z]) /. z -> {z +}), {z, -2 - 2 I, 2 + 2 I}]

this next snip is from Examples in Help for Binomial

ComplexPlot3D[Binomial[z, 1/5], {z, -2 - 2 I, 2 + 2 I}, 
 PlotLegends -> Automatic]

MathematicalFunctionData["Binomial", "NamedIdentities"] // Length == 18
{"Central binomial coefficient", "Eulerian number", ...}

(Mathematica has 18 Binomial theorems which can be shown, if you would like to see them)

I'm unsure about your objective. I don't doubt you have a modified binomial theorem that will apply for complex numbers - there are many theorems to appeal to. But I'm unsure if you are attempting to apply the theorem to find coefficients of the equations (the coefficients are already provided?). I'm unsure if the 7 equations are meant to be combined to prove a binomial theorem.

The notebook contains many syntax errors. You will need to use Help and follow examples or post the text of the theorem you wish to use.

I provided an example plot from the 7th order equation in the notebook.

Attachments:

complex.jpg