 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019-resit/2/solution

Here

i^{3} = - i

, so the argument condition concerns

z + i

z = x + i y

x + i (y + 1)

must point along the direction

1 + i

. Consequently

z = - i + s (1 + i), s > 0, equivalently y = x - 1, x > 0.

This is an open ray in the Argand plane, rather than the whole straight line. Its endpoint

z = - i

is excluded because the argument of zero is undefined; the opposite half-line has a different argument.

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