Some Words on Graphing Complex Functions
We know how to draw a complex number \( z = x + iy \) as \(x\) runs on the real axis (horizontal) and \(y\) runs on the imaginary axis (vertical). But how do we draw a complex function \(f(z)\)? A complex function transforms a number \(z = x + iy\) into another complex number \(w = f(z) = u + iv\). So we require four dimensions which is not possible for a human to visualize(maybe angels can do it). So we have two planes w-plane and z-plane. The z-plane is the domain of the function \(f\) and the w-plane is the range of the function \(f\).
For example the function $$w = z + 1 + i$$ $$u + iv = x + iy + 1 + i$$ $$u = x + 1, v = y + 1$$ Consider a straight line \(y=x\) then \[ u = x+1 \ and \ v = x+1, u = v \]
The graphs show that when y = x then u = v the line just translated by 1 unit in both directions, preserving the angle of \(\frac{\pi}{4}\), for y = x. Remembering the angle is \(\theta = \tan^{-1}(\frac{y}{x})\), which is the derivative.
Therefore, this particular mapping (transformation) is conformal.