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Complex numbers/Conjugation/Real part imaginary part/Fact/Proof/Exercise
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Complex numbers
|
Conjugation/Real part imaginary part/Fact
Show that for a
complex number
z
{\displaystyle {}z}
the following relations hold.
z
¯
=
Re
(
z
)
−
i
Im
(
z
)
{\displaystyle {}{\overline {z}}=\operatorname {Re} \,{\left(z\right)}-{\mathrm {i} }\operatorname {Im} \,{\left(z\right)}}
.
Re
(
z
)
=
z
+
z
¯
2
{\displaystyle {}\operatorname {Re} \,{\left(z\right)}={\frac {z+{\overline {z}}}{2}}}
.
Im
(
z
)
=
z
−
z
¯
2
i
{\displaystyle {}\operatorname {Im} \,{\left(z\right)}={\frac {z-{\overline {z}}}{2{\mathrm {i} }}}}
.
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