Solved Problem on Contours

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d) z = t + \frac{2 i}{t}, - \infty < t < 0

The function z is a parametric function of the type

z (t) = x (t) + i y (t)

Identifying the functions x(t) and y(t)

\begin{aligned} (I) & x (t) = t \\ (II) & y (t) = \frac{2}{t} \end{aligned}

substituting expression (I) into expression (II)

y = \frac{2}{x}

Graph 1

The function z(t) represents a branch of hyperbola oriented from (−∞, 0) to (0, −∞), (Graph 1).