CVP5

From Example Problems
Jump to: navigation, search

Show that \overline {z_{1}z_{2}}=\overline {z_{1}}\,\overline {z_{2}}\,

Let z_{1}=x_{1}+iy_{1}\, and z_{2}=x_{2}+iy_{2}\,. Then

\overline {z_{1}z_{2}} =\overline {(x_{1}+iy_{1})(x_{2}+iy_{2})}
=\overline {x_{1}x_{2}+x_{1}y_{2}i+x_{2}y_{1}i-y_{1}y_{2}}
=\overline {(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+x_{2}y_{1})}
=(x_{1}x_{2}-y_{1}y_{2})-i(x_{1}y_{2}+x_{2}y_{1})\,
=x_{1}x_{2}-x_{1}y_{2}i-x_{2}y_{1}i-y_{1}y_{2}\,
=(x_{1}-y_{1}i)(x_{2}-y_{2}i)\,
=\overline {(x_{1}+y_{1}i)}\,\overline {(x_{2}+y_{2}i)}
=\overline {z_{1}}\,\overline {z_{2}}


Main Page : Complex Variables : Proofs