Word Count : 485

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We are given the mapping: \[ w = f(z) = \frac{z - i}{z + i} \] and we need to determine the image of the circle: \[ |z| = r, \quad (r \neq 1) \] under this transformation. We will also examine what happens when \( r = 1 \). --- ### Step 1: Parameterizing the Circle A point on the circle \( |z| = r \) can be written as: \[ z = r e^{i\theta} = r(\cos\theta + i\sin\theta), \quad \theta \in [0, 2\pi] \] Substituting this into the given transformation: \[ w = \frac{z - __________ ____ _____ __________ _________ ____ __________ __________ ________ ________.
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