Solved Problem on Contours

\( \mathsf{a)}\;\; \displaystyle z=3t+it^{2}\qquad ,\qquad -\infty \lt t \lt \infty \)

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)

\[ \begin{align} & x(t)=3t \tag{I}\\[10pt] & y(t)=t^{2} \tag{II} \end{align} \]

using expression (I)

\[ \begin{gather} t=\frac{x}{3} \tag{III} \end{gather} \]

substituting expression (III) into expression (II)

\[ \begin{gather} y=\left(\frac{x}{3}\right)^{2}\\ y=\frac{x^{2}}{9} \end{gather} \]

For t = −1, we have x = 3×(−1) = −3 and y = (−1)² = 1, for t = 0, we have x = 0 and y = 0² = 0, for t = 1, we have x = 3×1 = 3 and y = 1² = 1, and so on.

Graph 1

The function z(t) represents a parabola oriented from −∞ to +∞, (Graph 1).

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