complex analysis branches and derivatives of Logarithms show

complex analysis: branches and derivatives of Logarithms

show that Log(i³) does not equal to 3 Log(i)

Solution

Log(i^3) = ln |(i^3)| + i Arg (i^3) = ln |-i| + i Arg (-i) = ln |sqrt 1| - i(pi/2) = (-pi/2)i

3 Log (i) = 3 [ ln | i | + i Arg (i)] = 3 [ ln (sqrt 1) + i ( pi/2)] = 3 [ (pi/2) i ] = (3pi/2)i

Note that -pi/2 and 3pi/2 are not equal, although they are 2 pi units apart

Hence, Log(i³) does not equal to 3 Log(i)