Practice Questions for CSIR NET Complex Analysis : Meromorphic Functions II

11. If \(f\) is entire \(|f(z)| \leq A|z|^{n}\) for some positive integers \(n\) and then

(a.) \(f(z)=z^{n}\)
(b.) \(f(z)=e^{n z}\)
(c.) \(f(z)\) is a polynomial of degree less than or equal to \(n\)
(d.) \(f(z)=n e^{z}\)

12. Let \(f\) be an entire function on \(\mathbb{C}\) such that \(|f(z)| \leq 100 \log |z|\) for each \(z\) with \(|z| \geq 2\). If \(f(i)=2i\), then \(f(1)\)

(a.) Must be 2
(b.) Must be \(2i\)
(c.) Must be \(i\)
(d.) Cannot be determined from the given data

13. Let \(f(z)\) be an entire function such that \(|f(z)| \leq K|z|\) for all \(z \in \mathbb{C}\), for some \(K>0\). If \(f(1)=i\), the value of \(f(i)\) is

(a.) 1
(b.) -1
(c.) \(i\)
(d.) \(-i\)

14. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be an arbitrary analytic function satisfying \(f(0)=0\) and \(f(1)=2\). Choose incorrect:

(a.) There exists a unique sequence \(\{z_{n}\}\) such that \(|z_{n}|>n\) and \(|f(z_{n})|>n\)
(b.) There exists a sequence \(\{z_{n}\}\) such that \(|z_{n}|>n\) and \(|f(z_{n})|
(c.) There exists a bounded sequence \(\{z_{n}\}\) such that \(|f(z_{n})|>n\)
(d.) There exists a sequence \(\{z_{n}\}\) such that \(z_{n} \rightarrow 0\) and \(f(z_{n}) \rightarrow 2\)

15. Let \(f(z)\) be an entire function such that for some constant \(\alpha, |f(z)| \leq \alpha|z|^{3}\) for \(|z| \geq 1\) and \(f(z)=f(iz)\) for all \(z \in \mathbb{C}\). Then

(a.) \(f(z)=\alpha z^{3}\) for all \(z \in \mathbb{C}\)
(b.) \(f(z)\) is constant.
(c.) \(f(z)\) is a quadratic polynomial.
(d.) No such \(f(z)\) exists.

16. Let \(B\) be an open subset of \(\mathbb{C}\) and \(\partial B\) denote the boundary of \(B\). Which of the following statements are correct?

(a.) For every entire function \(f\), we have \(\partial(f(B)) \subseteq f(\partial B)\)
(b.) For every entire function \(f\) and a bounded open set \(B\), we have \(\partial(f(B)) \subseteq f(\partial B)\)
(c.) For every entire function \(f\), we have \(\partial(f(B))=f(\partial B)\)
(d.) There exists an unbounded open subset \(B\) of \(\mathbb{C}\) and an entire function \(f\) such that \(\partial(f(B)) \subseteq f(\partial B)\)

17. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a meromorphic function analytic at 0 satisfying \(f\left(\frac{1}{n}\right)=\frac{n}{2n+1}\) for \(n \geq 1\)

(a.) \(f(0)=\frac{1}{2}\)
(b.) \(f\) has a simple pole at \(z=-2\)
(c.) \(f(2)=\frac{1}{4}\)
(d.) No such meromorphic function exists

18. Let \(f\) be a polynomial function on the entire complex plane such that \(f(z) \neq 0\) for \(z\) with \(|z|=1\). Then \(\frac{1}{2 \pi i} \int_{|z|=1} \frac{f^{\prime}(z)}{f(z)} dz\)

(a.) Can take only integer values
(b.) Can take only rational values
(c.) Is zero
(d.) Is equal to the degree of \(f\)

19. If \(f: \mathbb{C} \rightarrow \mathbb{C}\) is analytic and real-valued then

(a.) \(f\) is non-constant
(b.) \(f\) maps open sets to open sets
(c.) \(f\) is constant
(d.) \(f(z)=z^{n}\) for some \(n>0\)

20. Suppose \(f(z)=(z-1)(z-2)(z-3)^{3}\) and \(\Upsilon(t)=4 e^{i t}, 0 \leq t \leq 2 \pi\). Then the argument principle implies that \(\int_{\gamma} \frac{f^{\prime}(z)}{f(z)} dz\) is

(a.) \(2 \pi i\)
(b.) \(-2 \pi i\)
(c.) 0
(d.) \(10 \pi i\)