Practice Questions for CSIR NET Complex Analysis : Power Series and ROC

11. Consider the function \(f(z)=z^{2}(1-\cos z)\), \(z \in \mathbb{C}\). Which of the following are correct?

(a.) The function \(f\) has zeros of order 2 at 0.
(b.) The function \(f\) has zeros of order 1 at \(2 \pi n\), \(n= \pm 1, \pm 2, \ldots\).
(c.) The function \(f\) has zeros of order 4 at 0.
(d.) The function \(f\) has zeros of order 2 at \(-2 \pi n\), \(n= \pm 1, \pm 2, \ldots\).

12. Consider the power series \(\sum_{n=1}^{\infty} z^{n !}\). The radius of convergence of this series is

(a.) 0.
(b.) \(\infty\).
(c.) 1.
(d.) A real number greater than 1.

13. Which of the following functions \(f\) are entire functions and have simple zeros at \(z=i k\) for all \(k \in \mathbb{Z}\)?

(a.) \(f(z)=a_{n} z^{n}+a_{n-1} z^{n-1}+\ldots+a_{0}\) for some \(n \geq 1\) and some \(a_{0}, a_{1}, \ldots, a_{n} \in \mathbb{C}\).
(b.) \(f(z)=a \sin 2 \pi i z\), for some \(a \in \mathbb{C} \backslash\{0\}\).
(c.) \(f(z)=b \cos 2 \pi\left(i z-\frac{1}{4}\right)\), for some \(b \in \mathbb{C} \backslash\{0\}\).
(d.) \(f(z)=e^{c z}\), for some \(c \in \mathbb{C}\).

14. The power series \(\sum_{n=0}^{\infty} 3^{-n}(z-1)^{2 n}\) converges if

(a.) \(|z| \leq 3\).
(b.) \(|z|<\sqrt{3}\).
(c.) \(|z-1|<\sqrt{3}\).
(d.) \(|z-1| \leq \sqrt{3}\).

15. At \(z=0\), the function \(f(z) \stackrel{\vdots}{=} \exp \left(\frac{\zeta}{1-\cos z}\right)\) has

(a.) a removable singularity.
(b.) a pole.
(c.) an essential singularity.
(d.) The Laurent expansion of \(f(z)\) around \(z=0\) has infinitely many positive and negative powers of \(z\).

16. Consider the power series \(\sum_{n \geq 1} a_{n} z^{n}\) where \(a_{n}=\) number of divisors of \(n^{50}\). Then the radius of convergence of \(\sum_{n \geq 1} a_{n} z^{n}\) is

(a.) 1.
(b.) 50.
(c.) \(\frac{1}{50}\).
(d.) 0.

17. Let \(D=\{z \in \mathbb{C}:|z|<1\}\) and let \(f_{n}: D \rightarrow \mathbb{C}\) be defined by \(f_{n}(z)=\frac{z^{n}}{n}\) for \(n=1,2, \ldots\)

Then

(a.) The sequence \(\left\{f_{n}(z)\right\}\) and \(\left\{f_{n}^{\prime}(z)\right\}\) converge uniformly on \(D\).
(b.) The series \(\sum_{n=1}^{\infty} f_{n}(z)\) converges uniformly on \(D\).
(c.) The series \(\sum_{n=1}^{\infty} f_{n}^{\prime}(z)\) converges for each \(z \in D\).
(d.) The sequence \(\left\{f_{n}^{\prime \prime}(z)\right\}\) does not converge unless \(z=0\).

18. Let \(f, g\) be holomorphic functions defined as \(A \cup D\), where

\(A=\left\{z \in \mathbb{C}: \frac{1}{2}<|z|<1\right\}\) and

\(D=\{z \in \mathbb{C}:|z-2|<1\}\)

Which of the following statements are correct?

(a.) If \(f(z) g(z)=0\) for all \(z \in A \cup D\), then either \(f(z)=0\) for all \(z \in A\) or \(g(z)=0\) for all \(z \in A\).
(b.) If \(f(z) g(z)=0\) for all \(z \in D\), then either \(f(z)=0\) for all \(z \in D\) or \(g(z)=0\) for all \(z \in D\).
(c.) If \(f(z) g(z)=0\) for all \(z \in A\), then either \(f(z)=0\) for all \(z \in A\) or \(g(z)=0\) for all \(z \in A\).
(d.) If \(f(z) g(z)=0\) for all \(z \in A \cup D\), then either \(f(z)=0\) for all \(z \in A \cup D\), or \(g(z)=0\) for all \(z \in A \cup D\).

19. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be an entire function and let \(g: \mathbb{C} \rightarrow \mathbb{C}\) be defined by \(g(z)=f(z)-f(z+1)\) for \(z \in \mathbb{C}\). Which of the following statements are true?

(a.) If \(f\left(\frac{1}{n}\right)=0\) for all positive integers \(n\), then \(f\) is a constant function.
(b.) If \(f(n)=0\) for all positive integers \(n\), then \(f\) is a constant function.
(c.) If \(f\left(\frac{1}{n}\right)=f\left(\frac{1}{n}+1\right)\) for all positive integers \(n\), then \(g\) is a constant function.
(d.) If \(f(n)=f(n+1)\) for all positive integers \(n\), then \(g\) is a constant function.

20. The power series \(\sum_{0}^{\infty} 2^{-n} z^{2 n}\) converges if

(a.) \(|z| \leq 2\).
(b.) \(|z|<2\).
(c.) \(|z| \geq \sqrt{2}\).
(d.) \(|z|<\sqrt{2}\).

21. Let \(\mathbf{D}=\{z \in \mathbb{C}:|z|<1\}\) be the unit disc. Let \(f: \mathbf{D} \rightarrow \mathbb{C}\) be an analytic function satisfying \(f\left(\frac{1}{n}\right)=\frac{2 n}{3 n+1}\) for \(n \geq 1\). Then

(a.) \(f(0)=\frac{2}{3}\)
(b.) \(f\) has a simple pole at \(z=-3\)
(c.) \(f(3)=\frac{1}{3}\)
(d.) No such \(f\) exists.

22. At \(z=0\), the function \(f(z)=\frac{e^{z}+1}{e^{z}-1}\) has

(a.) A removable singularity
(b.) A pole
(c.) An essential singularity
(d.) The residue of \(f(z)\) at \(z=0\) is 2.

23. The poles of the function \(\frac{\cos z}{z^{2}+1}\) are

(a.) Integral multiples of \(2 \pi\)
(b.) Real
(c.) \(i\) and \(-i\)
(d.) \(\pm 1\)

24. Let \(r\) be the radius of convergence of \(\sum_{n=0}^{\infty} a_{n} z^{n}\). Let \(b_{n}=a_{n+2}\) for all \(n\). Then the radius of convergence of \(\sum b_{n} z^{n}\) is

(a.) \(r+2\)
(b.) \(r\)
(c.) \(r^{2}\)
(d.) \(r-2\)

25. In the Laurent series of \(\frac{\sin z}{z^{2}}\) at \(z=0\), the coefficient of \(\frac{1}{z}\) is

(a.) Zero
(b.) \(\pi\)
(c.) -1
(d.) 1