# B Simplifying the factors of a complex number's imaginary part

1. May 30, 2016

My question boils down to wondering if there is a way to simplify the imaginary part of a complex-valued function composed of n factors if the real and imaginary component for each of the factors is known but the factors may take on the value of their conjugate as well.

For example, is there a known way to simplify:

$Q = \Im (A^* \times B \times C^* \times D^*)$

where A, B, C, and D are each complex numbers themselves with known real and imaginary parts, $\Im$ is taking the imaginary part, and $^*$ corresponds to taking the complex conjugate of the complex number. Ideally, I'm wondering if there is known a way to simplify Q into another equivalent form such as

$Q = \Im (A) \Im (B^* \times C \times D)$.

I've attached my attempt at simplifying the imaginary part of a complex number composed of 3 factors that are themselves each complex-valued. It does feel like there is some kind of pattern but I'm not seeing and am oblivious to how it extends for larger systems and for complex conjugates taken at arbitrary positions. Would any of you happen to know if there's a known way to factorize such numbers?

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• ###### Screen Shot 2016-05-30 at 5.18.08 PM.png
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2. May 30, 2016

### Staff: Mentor

There is no formula that would make it easier. You can write all numbers as $A=A_R+iA_I$ and so on and then make a long list of summands that contribute to the imaginary part, e. g. $A_RB_RC_RD_I + A_RB_RC_ID_R + \dots$ (in total 8 components) but that is not very useful. If you know the numbers, where is the problem with just multiplying them?
Complex conjugation just changes the sign of the imaginary part, apart from that nothing happens so it is easy to take into account.

In your attached screenshot, two summands with c2 got lost.

3. May 30, 2016

One suggestion is to write each complex number as an amplitude and phase factor. i.e. $A=A_o e^{i \phi_A}$ etc. To get the final imaginary part, that is just a product of the amplitudes, $A_o B_o C_o D_o$, etc., multiplied by $\sin{\phi_{total}}$. (using Euler's formula). ($\phi_{total}$ is the sum of the phase factors.) Perhaps this is what you are looking for? (Note: A complex conjugate simply puts a minus sign on the phase of the number.)