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Mathematics
Calculus
How to know if a complex root is inside the unit circle
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[QUOTE="fresh_42, post: 6215361, member: 572553"] That was the idea, but I cannot see what you have done. ##z## is confusing here, as you use the same letter for different numbers. You haven't mentioned the quadratic equation up to now, only the solutions. But if ## z^4 + 2ikz^2 - 1 = 0 ## is your equation, I get ##z_{1,2,3,4} = \pm \sqrt{-ik \pm \sqrt{1-k^2}}##. I see what you have done to write it as you did. However, one has to be careful with complex numbers. Not every rule which is valid in ##\mathbb{R}## is also valid in ##\mathbb{C}##. See: [URL]https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/[/URL] I would therefore check the following equations, just to be sure: \begin{align*} &z^4+2ikz^2-1=\\ &=\left( z- \sqrt{-ik -\sqrt{1-k^2}} \right)\left(- \sqrt{-ik + \sqrt{1-k^2}} \right)\left( \sqrt{-ik - \sqrt{1-k^2}} \right)\left( \sqrt{-ik + \sqrt{1-k^2}} \right)\\ &=\left( z^2- i\left( \sqrt{k^2-1}-k\right) \right)\left(z^2-i\left(-\sqrt{k^2-1}-k\right) \right) \end{align*} Set ##\alpha=i\left( \sqrt{k^2-1}-k \right)\, , \,\beta=i\left( -\sqrt{k^2-1}-k \right)## and then we have ##\alpha \cdot \beta = - \left( -\sqrt{k^2-1}^2 +k^2 \right)= -1## and then what you wrote. [/QUOTE]
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How to know if a complex root is inside the unit circle
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