Practice Questions for CSIR NET Complex Analysis : Complex Integration and Singularities II

11. \(\int_{|z+1|=2} \frac{z^{2}}{4-z^{2}} d z=\)

(a.) 0.
(b.) \(-2 \pi i\).
(c.) \(2 \pi i\).
(d.) 1.

12. Let \(\gamma_{k}=\left\{k e^{i k \theta}: 0 \leq \theta \leq 2 \pi\right\}\) for \(k=1,2,3\). Which of the following are necessarily correct?

(a.) \(\frac{1}{2 \pi i} \int_{\gamma_{k}} \frac{1}{z} d z=0\) for \(k=1,2,3\).
(b.) \(\frac{1}{2 \pi i} \int_{\gamma_{1}} \frac{1}{z} d z=1\).
(c.) \(\frac{1}{2 \pi i} \int_{\gamma_{2}} \frac{1}{z} d z=4\).
(d.) \(\frac{1}{2 \pi i} \int_{\gamma_{3}} \frac{1}{z} d z=3\).

13. Let \(I_{r}=\int_{c_{r}} \frac{d z}{z(z-1)(z-2)}\), where \(C_{r}=\{z \in \mathbb{C}:|z|=r\}, r>0\). Then

(a.) \(I_{r}=2 \pi i\) if \(r \in(2,3)\).
(b.) \(I_{r}=\frac{1}{2}\) if \(r \in(0,1)\).
(c.) \(I_{r}=-2 \pi i\) if \(r \in(1,2)\).
(d.) \(I_{r}=0\) if \(r>3\).

14. Let \(f(z)=\frac{\cos (2 \pi z)}{z^{2}+1}\) and \(C=\left\{z:\left|z-\frac{1}{2} i\right|=\frac{1}{4}\right\}\) then \(\int_{C} f(z) d z\) is equal to

(a.) zero.
(b.) \(\pi i\).
(c.) \(-\pi i\).
(d.) \(2 \pi i\).

15. Let \(\gamma(t)=e^{i t}, 0 \leq t \leq 2 \pi\) then \(\frac{1}{2 \pi i} \int \frac{z^{2}}{\left(z-\frac{1}{2}\right)} d z\) is equal to

(a.) \(1 / 4\).
(b.) 2.
(c.) 4.
(d.) 0.

16. Let \(f(\grave{z})=\frac{z^{2}+3 z+2}{z^{2}-1}\) and \(C=\left\{z:\left|z-\frac{1}{2}\right|=1\right\}\) is then \(\int_{C} f(z) d z\) is equal to

(a.) zero.
(b.) \(2 \pi i\).
(c.) \(4 \pi i\).
(d.) \(6 \pi i\).

17. \(f: \mathbb{C} \rightarrow \mathbb{C}\) is a non-constant analytic function which of the following is possible?

(a.) \(\operatorname{Re} f(z)=1\) for all \(z\).
(b.) \(f(z)\) takes only integer values.
(c.) \(f(z)\) is an integer for every integer \(n\).
(d.) \(f(z)\) is bounded.

18. The contour integral \(\int_{C} \frac{\cos z}{z} d z\) where \(C\) is circle \(|z|=1\) is

(a.) \(2 \pi i\).
(b.) zero.
(c.) 1.
(d.) does not exist.

19. \(\int_{C} z^{2} d z\), where \(C\) is circle with centre 0 and radius 2 is equal.

(a.) \(4 \pi^{2}\).
(b.) \(2 \pi i\).
(c.) 4.
(d.) 0.

20. \(\Upsilon(t)=2 e^{i t}\) where \(t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) then \(\int_{\gamma} z^{2} d z=\) ?

(a.) \(-i\).
(b.) 0.
(c.) \(-\frac{16 i}{3}\).
(d.) \(1 / 2\).