Answer to Question #27103 in Complex Analysis for PARTHA SAHOO

Question

Show that Lagz can have real value if and only if < is positive

Answer 1

Recall that Log(z) is a multivalued function equal to
Log(z) = ln(|z|)+ i Agr(z), where Arg(z) is the argument of z.
Suppose for zome z
Log(z) = ln(|z|)+ i Agr(z) is real value.
This means that the imaginary part of Log(z) is zero, and so
Agr(z) = 0.
But this means that z belongs to the positive part of real axis, and so z is real and z&gt;0.
Conversely, suppose z&gt;0. Then Arg(z)=0, whence
Log(z) = ln(|z|) = ln(z) is real.