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August 18th, 2017, 09:43 AM   #1
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Best way to solve this complex equation

Hello everybody,

which is the best way to solve

(z* - 1/z)(z^2+8z*)=0 ?

(z* is z conjugate)

thank you
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August 18th, 2017, 10:28 AM   #2
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if the product equals zero then one or more of the factors must equal zero.

Solve that for both factors.

For example

$z^* - \dfrac 1 z = 0$

$z^* = \dfrac 1 z = \dfrac {z^*}{|z|^2}$

so $|z^2|=1$

and thus this is satisfied for any point on the unit circle.

You try the second factor.
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August 30th, 2017, 07:01 PM   #3
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Quote:
Originally Posted by romsek View Post
if the product equals zero then one or more of the factors must equal zero.
True for real numbers. You should show it's true for complex numbers- simple but not obvious.
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September 2nd, 2017, 08:49 AM   #4
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Quote:
Originally Posted by zylo View Post
True for real numbers. You should show it's true for complex numbers- simple but not obvious.
$\displaystyle z_{1}z_{2}=0 \rightarrow z_{1}$ and/or $\displaystyle z_{2}=0$ is not only obvious but trivial: z has an inverse if $\displaystyle z \neq 0$

My apologies.
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