91Ó°ÊÓ

Why is it important to study the fundamental theorem of algebra in a complex analysis course?

Suppose that \(f(z)=u(r, \theta)+i v(r, \theta)\) is analytic for all values of \(z=r e^{i \theta}\). Show that \(\int_{0}^{2 \pi}[u(r, \theta) \cos \theta-v(r, \theta) \sin \theta] d \theta=0\) Hint: Integrate \(f\) around the circle \(C_{1}^{+}(0)\).

Evaluate \(\int_{C}\left|z^{2}\right| d z\), where \(C\) is given by \(C: z(t)=t+i t^{2}\), for \(0 \leq t \leq 1\).

Evaluate \(\int_{C} \exp z \mathrm{~d} s\), where \(C\) is the straight-line segment joining 1 to \(1+i \pi\).

Find \(\int_{C_{3}^{+}(1)}\left(z^{2}+1\right)^{-2} d z\).

Evaluate \(\int_{C}^{z} \exp z d z\), where \(C\) is the square with vertices \(0,1,1+i\), and \(i\) taken with the counterclock wise orientation.

Show that \(\int_{C} 1 d z=z_{2}-z_{1}\), where \(C\) is the line segment from \(z_{1}\) to \(z_{2}\), by parametrizing \(C .\)

Evaluate \(\int_{C} \exp z d z\), where \(C\) is the straight-line segment joining 0 to \(1+i\).

Let \(P(z)=a_{0}+a_{1} z+a_{2} z^{2}+a_{3} z^{3}\). Find \(\int_{C_{1}^{+}(0)} P(z) z^{-n} d z\), where \(n\) is a positive integer.

Let \(z(t)=x(t)+i y(t)\), for \(a \leq t \leq b\), be a smooth curve. Give a meaning for each of the following expressions. (a) \(z^{\prime}(t)\). (b) \(\left|z^{\prime}(t)\right| d t\). (c) \(\int_{a}^{b} z^{\prime}(t) d t\) (d) \(\int_{a}^{b}\left|z^{\prime}(t)\right| d t\)