Complex is sometimes complex

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If z = x + iy stands for any complex number where i = 1 \sqrt{-1} .

The function f(x) = Log(z - i) is

entire singular at x = 0, y = 1 analytic everywhere except on the portion x x\leq 0 of the line y = 1 analytic everywhere except x > 0 of the line y = 1

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