The boundary imaginary axis passes through 0,−i,∞, so its image is the circle through 10,5−5i,5+5i: center 5, radius 5. Solving the map gives T(z)=(10−5iz)/(1−(1+i)z/2). Test z=−1 in the left half-plane: T(−1)=7+i, inside this circle. Therefore the left half-plane maps to the interior **C**.
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