Practice Questions for CSIR NET Complex Analysis : Meromorphic Functions I

1. Let \(p(z, w)=\alpha_{0}(z)+\alpha_{1}(z) w+\ldots .+\alpha_{k}(z) w^{k}\), where \(k \geq 1\) and \(\alpha_{0}, \ldots, \alpha_{k}\) are non-constant polynomials in the complex variable \(z\). Then \(\{(z, w) \in \mathbb{C} \times \mathbb{C}: p(z, w)=0\}\) is

(a.) Bounded with empty interior
(b.) Unbounded with empty interior
(c.) Bounded with nonempty interior
(d.) Unbounded with nonempty interior

2. \(f(z)\) is a single-valued complex function then

(a.) It is rational implies it is meromorphic
(b.) It is rational if it is meromorphic
(c.) It is meromorphic if and only if it is rational
(d.) Let \(z\) is a rational function as it has simple poles as its singularities

3. If \(\phi(z)=\operatorname{Re}(z)+f(z)\) where \(f(z)\) is meromorphic. Then

(a.) \(\phi(z)\) is meromorphic
(b.) \(\phi(z)\) and \(f(z)\) have the same number of singularities
(c.) \(\phi(z)\) is analytic in every closed and bounded region provided it has no poles
(d.) \(\phi(z)\) has no singularities

4. Select the incorrect statement

(a.) There doesn't exist an analytic function \(f\) in the neighborhood of 0 whose square is \(z\).
(b.) There doesn't exist a non-constant entire function \(f(z)\) such that \(e^{f(z)}\) is bounded
(c.) Every rational function is meromorphic
(d.) None of these

5. If \(M\) is the set of all meromorphic functions, \(R\) is the set of rational functions, and \(I\) is the set of functions having an essential singularity at infinity then

(a.) \(M \cap I\) is empty
(b.) \(R \cap I\) is empty
(c.) \(M \cap R\) is empty
(d.) Intersections of any pair of \(M\), \(R\), and \(I\) are non-empty

6. If \(f\) is a meromorphic function on \(C\) then

(a.) Poles of \(f\) are isolated but not zeroes
(b.) Zeroes of \(f\) are isolated but poles of \(f\) need not be isolated
(c.) Zeroes and poles of \(f\) are isolated
(d.) Neither poles nor zeroes of \(f\) are isolated

7. Let \(f, g\) be meromorphic functions on \(\mathbb{C}\). If \(f\) has a zero of order \(k\) at \(z=a\) and \(g\) has a pole of order \(m\) at \(z=a\) then \(g(f(z))\) has

(a.) zero of order \(km\) at \(z=a\)
(b.) pole of order \(k m\) at \(z=a\)
(c.) zero of order \(|k-m|\) at \(z=a\)
(d.) pole of order \(|k-m|\) at \(z=a\)

8. Let \(f\) be a meromorphic function on \(\mathbb{C}\) such that \(|f(z)| \geq|z|\) at each \(z\) where \(f\) is holomorphic. Then which of the following is/are true?

(a.) The hypotheses are contradictory, so no such \(f\) exists.
(b.) Such an \(f\) exists.
(c.) There is a unique \(f\) satisfying the given conditions.
(d.) There is an \(A \in \mathbb{C}\) with \(|A| \geq 1\) such that \(f(z)=A z\) for each \(z \in \mathbb{C}\).

9. Let \(f\) be an analytic function defined on \(\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}\) such that the range of \(f\) is confined in the set \(\mathbb{C} \backslash(-\infty, 0]\). Then

(a.) \(f\) is necessarily a constant function.
(b.) There exists an analytic function \(g\) on \(\mathbb{D}\) such that \(g(z)\) is a square root of \(f(z)\) for each \(z \in \mathbb{D}\).
(c.) There exists an analytic function \(g\) on \(\mathbb{D}\) such that \(\operatorname{Re} g(z) \geq 0\) and \(g(z)\) is a square root of \(f(z)\) for each \(z \in \mathbb{D}\).
(d.) There exists an analytic function \(g\) on \(\mathbb{D}\) such that \(\operatorname{Re} g(z) \leq 0\) and \(g(z)\) is a square root of \(f(x)\) for each \(z \in \mathbb{D}\).

10. Let \(f\) be an entire function such that \(\lim _{|z| \rightarrow \infty}|f(z)|=\infty\). Then

(a.) \(f\left(\frac{1}{z}\right)\) has an essential singularity at 0
(b.) \(f\) cannot be a polynomial
(c.) \(f\) has finitely many zeros
(d.) \(f\left(\frac{1}{z}\right)\) has a pole at 0