Complex conjugate variables as independent variables in polynomial equations

If you want to treat them as independent variables, you have to add the constraint ##z_1z_2=1##.In summary, the conversation discusses different ways to treat complex conjugate variables in polynomial equations as independent variables, including using polar form and introducing new variables. The correct method depends on the desired outcome and may require adding additional constraint equations.
  • #1
dm368
3
0
Hi,
Is there any trick to treat complex conjugate variables in polynomial equations as independent variables by adding some other constraint equation ? Say, we have polynomial equation $f(x,x^{*},y,...) = 0$. where x^{*} is the complex conjugate of variable $x$. I might think of taking $x = r e^{i \alpha} =z_{1}$ and $x^{*} = r e^{-i \alpha} = z_{2}$ i.e. in polar form and then taking the original equation $f(z_{1},z_{2},y,...) = 0$ intersecting with $z_{1} z_{2} = 1$. But I don't know if this is the correct way - I am missing something here, right ?

Thanks in advance,

Cheers,

dm368
 
Physics news on Phys.org
  • #2
This depends on what you want to achieve. Beside the polar form you mentioned, you can always introduce the variables ##\frac{1}{2}(x+x^*)\, , \,\frac{1}{2}(x-x^*)## which are purely real and purely imaginary.
 

1. What are complex conjugate variables?

Complex conjugate variables are pairs of numbers that have the same real parts but opposite imaginary parts. For example, the complex conjugate of 3+2i is 3-2i. In polynomial equations, complex conjugate variables are often used as independent variables to represent complex numbers.

2. Why are complex conjugate variables used as independent variables in polynomial equations?

Complex conjugate variables can help simplify and solve polynomial equations that involve complex numbers. By treating the real and imaginary parts separately, complex conjugate variables can make it easier to manipulate and solve equations.

3. How do complex conjugate variables relate to the complex plane?

In the complex plane, complex conjugate variables are represented by points that are mirror images of each other across the real axis. This is because the real parts of complex conjugate variables are the same, but the imaginary parts are opposite. This relationship is useful for visualizing and understanding complex numbers.

4. Can complex conjugate variables be used in any type of polynomial equation?

Yes, complex conjugate variables can be used in any type of polynomial equation, including quadratic, cubic, and higher-order polynomials. They are especially useful for equations that involve complex solutions.

5. Are there any limitations to using complex conjugate variables in polynomial equations?

While complex conjugate variables can be helpful in solving polynomial equations, they may not always be necessary. In some cases, it may be possible to simplify the equation and solve for real variables instead. Additionally, complex conjugate variables may not be appropriate for certain types of problems, such as those involving only real numbers.

Similar threads

Replies
0
Views
495
  • Topology and Analysis
Replies
2
Views
573
  • Linear and Abstract Algebra
Replies
8
Views
993
  • Linear and Abstract Algebra
Replies
1
Views
708
  • Calculus
Replies
3
Views
1K
Replies
1
Views
758
  • Linear and Abstract Algebra
Replies
4
Views
765
  • Linear and Abstract Algebra
Replies
2
Views
880
  • Linear and Abstract Algebra
Replies
17
Views
4K
Replies
6
Views
2K
Back
Top