Complex Logarithm Function

The function is w = az+b. The blue grid is in the z = x+iy domain, the red image is in the w = u+iv domain.

Consider w = log(z). Note that w = log(z) is a many valued expression. We define the function Log, the Principal Branch of the logarithm, w = Log(z) for z = reiq  by
Log(z) = log(|z|) + i arg(z)
= log(r) + iq
where -p < q ≤ p.

Remembering that z = reiq   will lead us to understand both the real axis and imaginary axis animations.

The choice of principal branch requires us not to cross the negative real axis.