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

41. Let \(f\) be a non-constant entire function. Which of the following properties is possible for \(f\) for each \(z \in \mathbb{C}\)?

(a.) \(\operatorname{Re} f(z)=\operatorname{Im} f(z)\).
(b.) \(|f(z)|<1\).
(c.) \(\operatorname{Im} f(z)<0\).
(d.) \(f(z) \neq 0\).

42. Consider the function \(f, g: \mathbb{C} \rightarrow \mathbb{C}\) defined by \(f(z)=e^{z}, \quad g(z)=e^{i z}\). Let \(S=\{z \in \mathbb{C}: \operatorname{Re} z \in[-\pi, \pi]\}\). Then

(a.) \(f\) is an onto entire function.
(b.) \(g\) is a bounded function on \(\mathbb{C}\).
(c.) \(f\) is bounded on \(S\).
(d.) \(g\) is bounded on \(S\).

43. Let \(f\) be an entire function. If \(\operatorname{Im} f \geq 0\), then

(a.) \(\operatorname{Re} f\) is constant.
(b.) \(f\) is constant.
(c.) \(f=0\).
(d.) \(f(z)=z, z \in D\).

44. Let \(f\) be an entire function. If \(\operatorname{Re} f\) is bounded then

(a.) Im \(f\) is constant.
(b.) \(f\) is constant.
(c.) \(f \equiv 0\).
(d.) \(f^{\prime}\) is a non-zero constant.

45. \(f\) is an entire function. If \(f\) satisfies the following two equations \(f(z+1)=f(z)\) and \(f(z+i)=f(z)\) for every \(z\) in \(\mathbb{C}\) then

(a.) \(f^{\prime}(z)=f(z)\).
(b.) \(f(z) \in \mathbb{R} \quad \forall z\).
(c.) \(f \equiv\) constant.
(d.) \(f\) is a non-constant polynomial.

46. Let \(p\) be a polynomial in one complex variable. Suppose all zeroes of \(p\) are in the upper half plane \(H=\{z \in \mathbb{C} \mid \operatorname{Im}(z)>0\}\). Then

(a.) \(\operatorname{Im} \frac{p^{\prime}(z)}{p(z)}>0\) for \(z \in \mathbb{R}\).
(b.) \(\operatorname{Re} i \frac{p^{\prime}(z)}{p(z)}<0\) for \(z \in \mathbb{R}\).
(c.) \(\operatorname{Im} \frac{p^{\prime}(z)}{p(z)}>0\) for \(z \in \mathbb{C}\), with \(\operatorname{Im} z<0\).
(d.) \(\operatorname{Im} \frac{p^{\prime}(z)}{p(z)}>0\) for \(z \in \mathbb{C}\), with \(\operatorname{Im} z>0\).

47. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be an entire function.

Suppose that \(f=u+i v\) where \(u, v\) are the real and imaginary parts of \(f\) respectively. Then \(f\) is constant if

(a.) \(\{u(x, y): z=x+i y \in \mathbb{C}\}\) is bounded.
(b.) \(\{v(x, y): z=x+i y \in \mathbb{C}\}\) is bounded.
(c.) \(\{u(x, y)+v(x, y): \mathrm{z}=x+i y \in \mathbb{C}\}\) is bounded.
(d.) \(\left\{u^{2}(x, y)+v^{2}(x, y): z=x+i y \in \mathbb{C}\right\}\) is bounded.

48. Let \(C\) be the circle \(|z|=3 / 2\) in the complex plane that is oriented in the counter clockwise direction. The value of \(a\) for which

\(\int_{c}\left(\frac{z+1}{z^{2}-3 z+2}+\frac{a}{z-1}\right) d z=0 \text { is }

(a.) 1.
(b.) -1.
(c.) 2.
(d.) -2.

49. The radius of convergence of the series \(\sum_{n=1}^{\infty} z^{n^{2}}\) is

(a.) 0.
(b.) \(\infty\).
(c.) 1.
(d.) 2.

50. Consider the following power series in the complex variable \(z\):

\(f(z)=\sum_{n=1}^{\infty} n \log n z^{n}, g(z)=\sum_{n=1}^{\infty} \frac{e^{n^{2}}}{n} z^{n}\). If \(r, R\) are the radii of convergence of \(f\) and \(g\) respectively, then

(a.) \(r=0, R=1\).
(b.) \(r=1, R=0\).
(c.) \(r=1, R=\infty\).
(d) \(r=\infty, R=1\).

51. \(\because f(z)=\tan \pi z\) and \(C:|z|=\pi\) then the value of \(\int_{C} \frac{f^{\prime}(z)}{f(z)} d z\) is

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