Principal square root of a product of complex numbers with positive real part Given n complex numbers z_i with ccRe z_i>0, why is it that prod_i sqrt(z_i)=sqrt(prod_i z_i)? Numerically, this appears to be the case, however, I don't see an easy way to prove it.

Brianna Schmidt

Brianna Schmidt

Answered question

2022-10-22

Principal square root of a product of complex numbers with positive real part
Given n complex numbers z i with z i > 0, why is it that
i z i = i z i ?
Numerically, this appears to be the case, however, I don't see an easy way to prove it.

Answer & Explanation

Adalyn Pitts

Adalyn Pitts

Beginner2022-10-23Added 15 answers

This is not true.
If your branch is defined over all z, with z > 0, but IT IS NOT defined at
z 0 = r e i ϑ , ϑ ( π / 2 , 3 π / 2 ) ,
then w = r 1 / 3 e i ϑ / 3 has positive real part, and your branch is not defined on z 0 = w w w

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