Chapter 18

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=z^{3}-1+3 i\)

Evaluate the given integral along the indicated contour. \(\int_{C}(z+3) d z\), where \(C\) is \(x=2 t, y=4 t-1,1 \leq t \leq 3\)

The sector defined by \(-\pi / 6<\arg z<\pi / 6\) is a simply connected domain.

\(\oint_{C} \frac{4}{z-3 i} d z ;|z|=5\)

Evaluate the given integral along the indicated contour. \(\int_{C}(2 \bar{z}-z) d z\), where \(C\) is \(x=-t, y=t^{2}+2,0 \leq t \leq 2\)

If \(\oint_{C} f(z) d z=0\) for every simple closed contour \(C\), then \(f\) is analytic within and on \(C\).

\(\oint_{C} \frac{z^{2}}{(z-3 i)^{2}} d z ;|z|=5\)

\(\int_{C}(2 \bar{z}-z) d z\), where \(C\) is \(x=-t, y=t^{2}+2,0 \leq t \leq 2\)

Evaluate the given integral along the indicated contour \(C\). \(\int_{C} 2 z d z\), where \(C\) is \(z(t)=2 t^{3}+i\left(t^{4}-4 t^{3}+2\right),-1 \leq t \leq 1\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\frac{z}{2 z+3}\)

The value of \(\int_{C} \frac{z-2}{z} d z\) is the same for any path \(C\) in the right half-plane \(\operatorname{Re}(z)>0\) between \(z=1+i\) and \(z=10+8 i .\)

\(\oint_{C} \frac{e^{z}}{z-\pi i} d z ;|z|=4\)

\(\int_{C} 2 z d z\), where \(C\) is \(z(t)=2 t^{3}+i\left(t^{4}-4 t^{3}+2\right),-1 \leq t \leq 1\)

\(\int_{C} z^{2} d z\), where \(C\) is \(z(t)=3 t+2 i t,-2 \leq t \leq 2\)

Evaluate the given integral along the indicated contour \(C\). \(\int_{C} 6 z^{2} d z\), where \(C\) is \(z(t)=2 \cos ^{3} \pi t-i \sin ^{2} \frac{\pi}{4} t, 0 \leq t \leq 2\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\frac{z-3}{z^{2}+2 z+2}\)

If \(g\) is entire, then \(\oint_{C} \frac{g(z)}{z-i} d z=\oint_{C_{1}} \frac{g(z)}{z-i} d z\), where \(C\) is the circle \(|z|=3\) and \(C_{1}\) is the ellipse \(x^{2}+y^{2} / 9=1\).

\(\oint_{C} \frac{1+2 e^{z}}{z} d z ;|z|=1\)

\(\int_{C}\left(3 z^{2}-2 z\right) d z\), where \(C\) is \(z(t)=t+i t^{2}, 0 \leq t \leq 1\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\frac{\sin z}{\left(z^{2}-25\right)\left(z^{2}+9\right)}\)

Evaluate the given integral along the indicated contour. \(\int_{C} \frac{1+z}{z} d z\), where \(C\) is the right half of the circle \(|z|=1\) from \(z=-i\) to \(z=i\)

\(\oint_{C} \frac{z^{2}-3 z+4 i}{z+2 i} d z ;|z|=3\)

\(\int_{0}^{3+i} z^{2} d z\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\frac{e^{z}}{2 z^{2}+11 z+15}\)

Evaluate the given integral along the indicated contour. . \(\int_{C}|z|^{2} d z\), where \(C\) is \(x=t^{2}, y=1 / t, 1 \leq t \leq 2\)

If \(f(z)=\oint_{C} \frac{\xi^{2}+6 \xi-2}{\xi-z} d \xi\), where \(C\) is \(|z|=3\), then \(f(1+i)=\)

\(\int_{-2 i}^{1}\left(3 z^{2}-4 z+5 i\right) d z\)

\(\int_{C}|z|^{2} d z\), where \(C\) is \(x=t^{2}, y=1 / t, 1 \leq t \leq 2\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\tan z\)

Evaluate the given integral along the indicated contour. \(\oint_{c} \operatorname{Re}(z) d z\), where \(C\) is the circle \(|z|=1\)

If \(f(z)=z^{3}+e^{z}\) and \(C\) is the contour \(z=8 e^{i t}, 0 \leq t \leq 2 \pi\), then \(\oint_{C} \frac{f(z)}{(z+\pi i)^{3}} d z=\)

\(\oint_{C} z^{2}+4 d z\); (a) \(|z-i|=2\), (b) \(|z+2 i|=1\)

\(f(z)=\tan z\)

\(\oint_{C} \operatorname{Re}(z) d z\), where \(C\) is the circle \(|z|=1\)

Prove that \(\oint_{c} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\). \(f(z)=\frac{z^{2}-9}{\cosh z}\)

Evaluate the given integral along the indicated contour. \(\oint_{c}\left(\frac{1}{(z+i)^{3}}-\frac{5}{z+i}+8\right) d z\), where \(C\) is the circle \(|z+i|=1\), \(0 \leq t \leq 2 \pi\)

If \(f\) is entire and \(|f(z)| \leq 10\) for all \(z\), then \(f(z)=\)

\(\int_{-3 i}^{2 i}\left(z^{3}-z\right) d z\)

\(\oint_{C}\left(\frac{1}{(z+i)^{3}}-\frac{5}{z+i}+8\right) d z\), where \(C\) is the circle \(|z+i|=1\), \(0 \leq t \leq 2 \pi\)

\(\oint_{C} \frac{z^{2}+4}{z^{2}-5 i z-4} d z ;|z-3 i|=1.3\)

\(\int_{-i / 2}^{1-i}(2 z+1)^{2} d z\)

\(\int_{C}\left(x^{2}+i y^{3}\right) d z\), where \(C\) is the straight line from \(z=1\) to \(z=i\)

If \(f\) is analytic within and on the simple closed contour \(C\) and \(z_{0}\) is a point within \(C\), then $$ \oint_{c} \frac{f^{\prime}(z)}{z-z_{0}} d z=\oint_{C} \frac{f(z)}{\left(z-z_{0}\right)^{2}} d z $$

\(\int_{1}^{i}(i z+1)^{3} d z\)

\(\int_{C}\left(x^{3}-i y^{3}\right) d z\), where \(C\) is the lower half of the circle \(|z|=1\) from \(z=-1\) to \(z=1\)

Use any of the results in this section to evaluate the given integral along the indicated closed contour(s). \(\oint_{C}\left(z+\frac{1}{z}\right) d z ;|z|=2\)

Evaluate the given integral along the indicated contour. \(\int_{C} e^{2} d z\), where \(C\) is the polygonal path consisting of the line segments from \(z=0\) to \(z=2\) and from \(z=2\) to \(z=1+\pi i\)

\(\oint_{C} \frac{e^{z^{2}}}{(z-i)^{3}} d z ;|z-i|=1\)

\(\oint_{C}\left(z+\frac{1}{z}\right) d z ;|z|=2\)

\(\int_{C} e^{2} d z\), where \(C\) is the polygonal path consisting of the line segments from \(z=0\) to \(z=2\) and from \(z=2\) to \(z=1+\pi i\)