Why Must ##\mu_1, \mu_2## = ##\mu_1^*, \mu_2^*##? Parametric Resonance

[arpon]

Why $\mu_1, \mu_2$ must be the same as $\mu_1^*, \mu_2^*$ ?

What I thought is : If $\mu_1\mu_2 = \mu_1^*\mu_2^*$ and $\mu_1+\mu_2 = \mu_1^*+\mu_2^*$, then $\mu_1, \mu_2$ are the same as $\mu_1^*, \mu_2^*$

It can be shown by taking the complex conjugate of (27.5) that $$\mu_1\mu_2 = \mu_1^*\mu_2^*=1$$
Now it is to be proven that $\mu_1+\mu_2 = \mu_1^*+\mu_2^*$.
Any help would be appreciated.

 

[scottdave]

If u1 = (a + bi) and u2 = (c + di), then you can write (a + bi)(c + di) = 1. Expanding it, you get:

ac + adi + bci – bd = 1. Rearrange as (ac – bd) + i(ad + bc) = 1. So (ac – bd) = 1, and (ad + bc) = 0. You may be able to get a starting point from that. If possible, I will try to come back and revisit this.

 

[arpon]

If u1 = (a + bi) and u2 = (c + di), then you can write (a + bi)(c + di) = 1. Expanding it, you get:

ac + adi + bci – bd = 1. Rearrange as (ac – bd) + i(ad + bc) = 1. So (ac – bd) = 1, and (ad + bc) = 0. You may be able to get a starting point from that. If possible, I will try to come back and revisit this.

Thanks for your help. But still could not get it.