Thread: complex log

Show that if z1= i and z2= i-1, then Log(z1z2) is not equal to Log(z1)+Log(z2).
I have that they are equal but I don't know what I did wrong.
Log(z1z2)= (1/2)*Log(2) + i(5Pi/4)
Log(z1) + Log(z2) = (1/2)*Log(2) + i(5Pi/4)

Originally Posted by frankdent1

Show that if z1= i and z2= i-1, then Log(z1z2) is not equal to Log(z1)+Log(z2).
I have that they are equal but I don't know what I did wrong.
Log(z1z2)= (1/2)*Log(2) + i(5Pi/4)
Log(z1) + Log(z2) = (1/2)*Log(2) + i(5Pi/4)

This depends on your definition of Log. The complex logarithm is a multi-valued function, and you obtain its principal value Log by restricting the imaginary part to lie in some interval of length 2π. If the imaginary part lies in the interval from 0 to 2π then Log(z1z2) and Log(z1) + Log(z2) are both equal to (1/2)Log(2) + i(5π/4). But it's more usual to define Log by restricting the imaginary part to the interval from –π to π. In that case, Log(z1) + Log(z2) would be the same as before, but you would have Log(z1z2) = (1/2)Log(2) – i(3π/4).