Fisicaexe - Solved Problems in Physics

Contours

Identify the curves or arcs of the equations given below, make the graphs and identify the directions in each case.

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

Solution

\( \mathsf{b)}\;\; \displaystyle z=r(\cos t+i\operatorname{sen}t)\qquad ,\qquad -\frac{\pi }{4}\leqslant t\leqslant \pi \qquad ,\qquad r\gt 0 \)

Solution

\( \mathsf{c)}\;\; \displaystyle z=\frac{1}{t}+it\qquad ,\qquad 1\leqslant t\lt \infty \)

Solution

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

Solution

\( \mathsf{e)}\;\; \displaystyle z=t+i\sqrt{1-t^{2}\;}\qquad ,\qquad -1\leqslant t\leqslant 1 \)

Solution

\( \mathsf{f)}\;\; \displaystyle z=t-i\sqrt{1-t^{2}\;}\qquad ,\qquad -1\leqslant t\leqslant 1 \)

Solution

\( \mathsf{g)}\;\; \displaystyle z=\sqrt{1-t^{2}\;}+it\qquad ,\qquad -1\leqslant t\leqslant 1 \)

Solution