Find the image of the circle  under the mapping  . What happens when ?

To find the image of the circle \(|z| = r\) under the mapping \(w = f(z) = \frac{z-i}{z+i}\), we substitute \(z = re^{i\theta}\) into the mapping function, where \(r > 0\) and \(0 \leq \theta < 2\pi\). 

So, \(f(z) = \frac{re^{i\theta} - i}{re^{i\theta} + ________ ____ _________ ____ ___ ______ ___.
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