Question #27103

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

Expert's answer

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>0.

Conversely, suppose z>0. Then Arg(z)=0, whence

Log(z) = ln(|z|) = ln(z) is real.

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>0.

Conversely, suppose z>0. Then Arg(z)=0, whence

Log(z) = ln(|z|) = ln(z) is real.

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