Solved Problem on Contours

\( \mathsf{d)}\;\; \displaystyle z=t+\frac{2i}{t}\qquad ,\qquad -\infty \lt t \lt 0 \)

The function z is a parametric function of the type

\[ \bbox[#99CCFF,10px] {z(t)=x(t)+iy(t)} \]

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

\[ \begin{align} & x(t)=t \tag{I}\\[10pt] & y(t)=\frac{2}{t} \tag{II} \end{align} \]

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).