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

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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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