Question

How to simplify
\((2\overline{z}-i)(2z+i) =|2z+i|^2\)
Please, help

Answer 1

nvm got it

Answer 2

let \[z=x+i y, \space \overline z=x-iy\]
\[\overline z \times z=x^2+y^2\]
left hand side:
\[(2\overline{z}-i)(2z+i)=[4(x^2+y^2)+2i(x-iy)-2i(x+iy)+1]\]
\[[4(x^2+y^2)+2i(x-iy)-2i(x+iy)+1]=[(4x^2+4y^2)+2ix+2y-2ix+2y+1]\]
\[[4x^2+4y^2)+2ix+2y-2ix+2y+1]=[(4x^2+4y^2)+4y+1]\]
right hand side:
\[|2z+i|^2=(\sqrt{(2x)^2+i(2y+1)^2}=4x^2+4y^2+4y+1\]
so LHS=RHS