|COMPLEX NUMBERS| Determine the complex conjugate of... ? Thx!

Question

$((a+bi)/(a -bi))^2 - ((a-bi)/(a +bi))^2$

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Answer 1 · 2018-01-10T15:24:02

See below.

If $z = x+i y$ then $bar z = x-iy$ so

$bar(((a+bi)/(a -bi))^2 - ((a-bi)/(a +bi))^2) = ((a-bi)/(a +bi))^2 - ((a+bi)/(a -bi))^2 = (8ab(b^2- a^2 ))/(a^2+b^2)^2i$

Answer 2 · 2018-01-11T04:16:12

Complex conjugate is $-(8ab(a^2-b^2)i)/((a^2+b^2)^2$

$((a+bi)/(a-bi))^2-((a-bi)/(a+bi))^2$

= $(a^2-b^2+2abi)/(a^2-b^2-2abi)-(a^2-b^2-2abi)/(a^2-b^2+2abi)$

= $((a^2-b^2)^2+4ab(a^2-b^2)i-4a^2b^2-((a^2-b^2)^2-4ab(a^2-b^2)i-4a^2b^2))/((a^2-b^2)^2+4a^2b^2)$

= $(8ab(a^2-b^2)i)/((a^2+b^2)^2$

Hence complex conjugate is $-(8ab(a^2-b^2)i)/((a^2+b^2)^2)$