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

31. Let \(\gamma\) be a simple closed curve in the complex plane then the set of all possible values of \(\oint_{\gamma} \frac{d z}{z(1-z^{2})}\) is

(a.) \(\{0, \pm \pi i\}\).
(b.) \(\{0, \pm \pi i, 2 \pi i\}\).
(c.) \(\{0, \pm \pi i, \pm 2 \pi i\}\).
(d.) \(\{0\}\).

32. For the positively oriented unit circle, \(\int_{|z|=1} \frac{2 \operatorname{Re}(z)}{z+2} d z\) equals -

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

33. Let \(\gamma\) be a curve given by \(r=2+4 \cos \theta\), \(0 \leq \theta \leq 2 \pi\). If \(I_{1}=\int_{\gamma} \frac{d z}{(z-1)}\) and \(I_{2}=\int_{\gamma} \frac{d z}{(z-3)}\), then

(a.) \(I_{1}=2 I_{2}\).
(b.) \(I_{1}=I_{2}\).
(c.) \(2 I_{1}=I_{2}\).
(d.) \(I_{1}=0, I_{2} \neq 0\).

34. Let \(I=\int_{C} \frac{\cot (\pi z)}{(z-i)^{2}} d z\), where \(C\) is the contour \(4 x^{2}+y^{2}=2\) (counter clockwise) then \(I\) is equal to

(a.) 0.
(b.) \(-2 \pi i\).
(c.) \(2 \pi i\left(\frac{\pi}{\sinh^{2}\pi}-\frac{1}{\pi}\right)\).
(d.) \(\frac{-2 \pi^{2} i}{\sinh^{2}\pi}\).

35. Let \(r\) be any circle enclosing the origin and oriented counter-clockwise. then the value of the integral \(\int_{\gamma} \frac{\cos z}{z^{2}} d z\) is

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

36. The value of the integral \(\int_{C} \frac{d z}{z^{2}-1}\) where \(C:|z|=4\) is equal to

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

37. The value of the integral \(\int_{C} \frac{\sin \pi z^{2}+\cos \pi z^{2}}{(z-4)(z-2)} d z\) where \(C\) is the circle \(|z|=3\) traced anticlockwise is

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

38. The value of \(\int_{|z|=2} \frac{e^{z}}{(z+1)^{2}} d z\) is

(a.) \(2 \pi i e^{-1}\).
(b.) \(\frac{8 \pi i}{3} e^{-6}\).
(c.) \(\frac{2 \pi i}{3} e^{-2}\).
(d.) 0.

39. Let \(f\) be an entire function. Which of the following statements are correct?

(a.) \(f\) is constant if the range of \(f\) is contained in a straight line.
(b.) \(f\) is constant if \(f\) has uncountably many zeros.
(c.) \(f\) is constant if \(f\) is bounded on \(\{z \in \mathbb{C}: \operatorname{Re}(z) \leq 0\}\).
(d.) \(f\) is constant if the real part of \(f\) is bounded.

40. Let \(f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}\) be an entire function and let \(r\) be a positive real number. then

(a.) \(\sum_{n=0}^{\infty}\left|a_{n}\right|^{2} r^{2 n} \leq \sup_{|z|=r}|f(z)|^{2}\).
(b.) \(\sup_{|z|=r}|f(z)|^{2} \leq \sum_{n=0}^{\infty}\left|a_{n}\right|^{2} r^{2 n}\).
(c.) \(\sum_{n=0}^{\infty}\left|a_{n}\right|^{2} r^{2 n}=\frac{1}{2 \pi} \int_{0}^{2 \pi}\left|f\left(r e^{i \theta}\right)\right|^{2} d \theta\).
(d.) \(\sup_{|z|=r}|f(z)|^{2} \leq \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|f\left(r e^{i \theta}\right)\right|^{2} d \theta\).