Regarding solving a polynomial: gives empty output

Questions
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1

Hello All,

I am trying to solve the equation given in the notebook attached.
The algebraic expression is not giving the roots. It is showing like, {{}}.
I don’t understand what does it mean. Can anyone help me this?
How should I solve such an expression?

Open in Wolfram Cloud ↗

Untitled-1.nb (498 KB)

2

The empty lists means that Solve found no solutions. As for how to find solutions, your expressions are way too complicated for me to try to debug your code.

3

Your equation is of degree 6 in omega, with very complicated coefficients:

In[318]:= PolynomialQ[Expand[xxx*\[Omega]^2], \[Omega]]
Exponent[Expand[xxx*\[Omega]^2], \[Omega]]

Out[318]= True

Out[319]= 6

It seems hopless, unless you set most parameters to numbers.

4

Even if I put the value of \Kappa = 2 (say), It is showing output as complex infinity.

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.

Later If I evaluate it element wise, it is showing complex infinity for

HypergeometricPFQ[{1/2, 1/2, 1}, {-\[Kappa] , 1, 1, 1}, 
 2*\[Lambda]*\[Kappa]]

and showing some answer for

HypergeometricPFQ[{1, 1/2, 1}, {-\[Kappa] + 1/2, 1, 1, 1}, 
 2*\[Lambda]*\[Kappa]]

What is the problem ?

hypergeometic.nb (8.53 KB)

5

Try with kappa=1/7:

xxx /. {me -> 1, Te -> 1, \[Omega]pe -> 1, \[Omega]ce -> 1, 
  ky -> 1, \[Epsilon]b -> 1, c -> 1, k -> 1, kpar -> 1, 
  kz -> 1, \[Omega]pi -> 1, 
  vde -> 1, \[Kappa] -> 1/7, \[Phi] -> Pi/6}
N[%*\[Omega]^2 // Simplify, 10] // Expand
6

There is a limit for [kappa] value, it should be greater than 3/2.

7

Ok, try with 3/2+1/5:

In[17]:= 
xxxx /. {me -> 1, Te -> 1, \[Omega]pe -> 1, \[Omega]ce -> 1, 
   ky -> 1, \[Epsilon]b -> 1, c -> 1, k -> 1, kpar -> 1, 
   kz -> 1, \[Omega]pi -> 1, 
   vde -> 1, \[Kappa] -> 3/2 + 1/5, \[Phi] -> Pi/6};
N[%*\[Omega]^2 // Simplify, 10] // Expand

Out[18]= -49.8649146 - 3491.27281 \[Omega] + 9421.90942 \[Omega]^2 + 
 643294.193 \[Omega]^3 - (42938.7191 + 141.5690054 I) \[Omega]^4 + 
 17009.04504 \[Omega]^6