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

21. If \(f\) and \(g\) are analytic on the same region \(D\) and if \(f(z) g(z)=0\) for all \(z\) in the region \(D\) then

(a.) at least one of \(f\) and \(g\) vanishes identically on \(D\).
(b.) \(f\) and \(g\) both vanish everywhere on \(D\).
(c.) \(f\) vanishes only at finitely many points of \(D\) and \(g\) never vanishes.
(d.) \(f\) and \(g\) vanish only at finitely many points of \(D\).

22. Let \(C=\{z \in \mathbb{C}:|z-i|=2\}\) Then \(\frac{1}{2 \pi} \oint_{C} \frac{z^{2}-4}{z^{2}+4} d z\) is equal to

(a.) Zero.
(b.) \(-2).
(c.) \(\frac{-\pi i}{2}\).
(d.) \(\frac{\pi i}{2}\).

23. Let \(\Omega=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}\) and let \(C\) be a smooth curve lying in \(\Omega\) with initial point \(-1+2 i\) and final point \(1+2 i\). The value of \(\int_{C} \frac{1+2 z}{1+z} d z\) is

(a.) \(4-\frac{1}{2} \ln 2+i \frac{\pi}{4}\).
(b.) \(-4+\frac{1}{2} \ln 2+i \frac{\pi}{4}\).
(c.) \(4+\frac{1}{2} \ln 2-i \frac{\pi}{4}\).
(d.) \(4-\frac{1}{2} \ln 2+i \frac{\pi}{2}\).

24. Let \(\int_{C}\left[\frac{1}{(z-2)^{4}}-\frac{(a-2)^{2}}{z}+4\right] d z=4 \pi\), where the closed curve \(C\) is the triangle having vertices at \(i,\left(\frac{-1-i}{\sqrt{2}}\right)\) and \(\left(\frac{1-i}{\sqrt{2}}\right)\),

(a.) \(1+i\).
(b.) \(2+i\).
(c.) \(3+i\).
(d.) \(4+i\).

25. Let \(r=\int_{\mathcal{C}} \frac{f(z)}{(z-1)(z-2)} d z\), \(f(z)=\sin \frac{\pi z}{2}+\cos \frac{\pi z}{2}\) and \(C\) is the curve \(|z|=3\) oriented anti-clockwise. Then the value of \(r\) is

(a.) \(4 \pi i\).
(b.) 0.
(c.) \(-2 \pi i\).
(d.) \(-4 \pi i\).

26. Let \(S\) be open unit disk and \(f: S \rightarrow \mathbb{C}\) be a real valued analytic function with \(f(0)=1\), then the set \(\{z \in S: f(z) \neq 1\}\) is

(a.) Empty.
(b.) Non-empty finite.
(c.) Countably infinite.
(d.) Uncountable.

27. Which one of the following does not hold for all continuous functions \(f:[-\pi, \pi] \rightarrow \mathbb{C}\)?

(a.) If \(f(-t)=f(t)\) for each \(t \in[-\pi, \pi]\) then \(\int_{-\pi}^{\pi} f(t) d t=2 \int_{0}^{\pi} f(t) d t\).
(b.) If \(f(-t)=-f(t)\) for each \(t \in[-\pi, \pi]\) then \(\int_{-\pi}^{\pi} f(t) d t=0\).
(c.) \(\int_{-\pi}^{\pi} f(-t) d t=-\int_{-\pi}^{\pi} f(t) d t\).
(d.) There is an \(\alpha\) with \(-\pi<\alpha<\pi\) such that \(\int_{-\pi}^{\pi} f(t) d t=2 \pi f(\alpha)\).

28. Let \(S\) be the positively oriented circle given by \(|z-2 i|=2\), then the value of \(\int_{S} \frac{d z}{z^{2}+4}\) is

(a.) \(-\frac{\pi}{2}\).
(b.) \(\frac{\pi}{2}\).
(c.) \(\frac{-i \pi}{2}\).
(d.) \(\frac{i \pi}{2}\).

29. Let \(f(z)\) be an analytic function, then the value of \(\int_{0}^{2 \pi} f\left(e^{i t}\right) \cos t d t\) equals

(a.) 0.
(b.) \(2 \pi f(0)\).
(c.) \(2 \pi f^{\prime}(0)\).
(d.) \(\pi f^{\prime}(0)\).

30. Define \(f: \mathbb{C} \rightarrow \mathbb{C}\) by \(f(z)=\begin{cases}0 & \text{if} \ \operatorname{Re}(z)=0 \\ z & \text{or} \ \operatorname{Im}(z)=0\end{cases}\) then the set of points where \(f\) is analytic is

(a.) \(\{z: \operatorname{Re}(z) \neq 0\) and \(\operatorname{Im}(z) \neq 0\}\).
(b.) \(\{z: \operatorname{Re}(z) \neq 0\}\).
(c.) \(\{z: \operatorname{Re}(z) \neq 0\) or \(\operatorname{Im}(z) \neq 0\}\).
(d.) \(\{z: \operatorname{Im}(z) \neq 0\}\).