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

1. For \(z \in \mathbb{C}\), define \(f(z)=\frac{e^{z}}{e^{z}-1}\). Then,

(a.) \(f\) is entire.
(b.) The only singularities of \(f\) are poles.
(c.) \(f\) has infinitely many poles on the imaginary axis.
(d.) Each pole of \(f\) is simple.

2. Let \(f(z)=\frac{z-1}{\exp \left(\frac{2 \pi i}{z}\right)-1}\). Then,

(a.) \(f\) has an isolated singularity at \(z=0\).
(b.) \(f\) has a removable singularity at \(z=1\).
(c.) \(f\) has infinitely many poles.
(d.) each pole of \(f\) is of order 1.

3. For the function \(f(z)=\sin \left(\frac{1}{\cos (1 / z)}\right)\), the point \(z=0\) is

(a.) a removable singularity.
(b.) a pole.
(c.) an essential singularity.
(d.) a non-isolated singularity.

4. An example of a function with a non-isolated essential singularity at \(z=2\) is

(a.) \(\tan \left(\frac{1}{\dot{z}-2}\right)\).
(b.) \(\sin \left(\frac{1}{z-2}\right)\).
(c.) \(e^{-(z-2)}\).
(d.) \(\tan \left(\frac{z-2}{z}\right)\).

5. For the function \(f(z)=\sin \left(\frac{1}{z}\right), z=0\) is a

(a.) Removable singularity.
(b.) simple pole.
(c.) non-isolated singularity.
(d.) Essential singularity.

6. If \(f(z)=z^{3}\) then it

(a.) has an essential singularity at \(z=\infty\).
(b.) has a pole of order 3 at \(z=\infty\).
(c.) has a pole of order 3 at \(z=0\).
(d.) is analytic at \(z=\infty\).

7. Let \(f(z)=\frac{e^{\frac{c}{(z-a)}}}{e^{\frac{z}{a}}-1}\).

(a.) \(z=0\) is a removable singularity.
(b.) \(z=a\) is an isolated essential singularity.
(c.) \(z=a\) is a non-isolated singularity.
(d.) \(z=0\) is a pole of order 2.

8. Let \(f\) be an entire function on \(\mathbb{C}\). Let \(g(z)=\overline{f(\bar{z})}\).

(a.) If \(f(z) \in \mathbb{R}\) for all \(z \in \mathbb{R}\) then \(f=g\).
(b.) If \(f(z) \in \mathbb{R}\) for all \(z \in\{z \mid \operatorname{Im} z=0\} \cup\{z \mid \operatorname{Im} z=a\}\), for some \(a>0\), then \(f(z+i a)=f(z-i a)\) for all \(z \in \mathbb{C}\).
(c.) If \(f(z) \in \mathbb{R}\) for all \(z \in\{z \mid \operatorname{Im} z=0\} \cup\{z \mid \operatorname{Im} z=a\}\), for some \(a>0\), then \(f(z+i a)=f(z)\) for all \(z \in \mathbb{C}\).
(d.) If \(f(z) \in \mathbb{R}\) for all \(z \in\{z \mid \operatorname{Im} z=0\} \cup\{z \mid \operatorname{Im} z=a\}\), for some \(a>0\), then \(f(z+2 i a)=f(z)\) for all \(z \in \mathbb{C}\).

9. Which of the following is incorrect about \(f\) where \(f(z)=\frac{z \cos (\pi z / 2 a)}{(z-a)\left(z^{2}+b^{2}\right)^{7} \sin ^{5} z}\)

(a.) \(f\) has removable singularities at \(z=a\).
(b.) \(f\) has poles of order 7 at \(z= \pm b i\).
(c.) \(f\) has poles of order 5 at \(z=0\).
(d.) \(f\) has poles of order 5 at \(z=k \pi, k= \pm 1, \pm 2, \ldots\).

10. Let \(u(x, y)\) be the real part of an entire function \(f(z)=u(x, y)+i v(x, y)\) for \(z=x+i y \in \mathbb{C}\). If \(C\) is the positively oriented boundary of a rectangular region \(R\) in \(\mathbb{R}^{2}\), then \(\oint_{C}\left[\frac{\partial u}{\partial y} d x-\frac{\partial u}{\partial x} d y\right]=\)

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