Complex Logarithm Function
The function is w = az+b. The red grid is in the
z = x+iy domain, the blue 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.