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

26. The function \(\frac{\sin z}{z^{2}}\) has

(a.) A pole of order 2 at the origin with residue 1.
(b.) A pole of order 1 at the origin with residue 1.
(c.) A pole of order 1 at the origin with residue 2.
(d.) An essential singularity at the origin with residue 1.

27. \(f(z)=\frac{\sin z}{z^{2}}\) has

(a) No singularities
(b.) An essential singularity at \(z=0\)
(c.) A removable singularity at \(z=0\)
(d.) A pole of order 1 at \(z=0\)

28. The coefficient of \(\frac{1}{z}\) in the Laurent series of \(\frac{\sin 2 z}{z^{2}}\) is

(a.) 0
(b.) 2
(c.) -1
(d.) 1

29. An example of a power series whose radius of convergence is 0 is

(a.) \(\sum_{n=0}^{\infty} z^{n}\)
(b.) \(\sum_{n=0}^{\infty} n z^{n}\)
(c.) \(\sum_{n=0}^{\infty} \frac{1}{z^{n}}\)
(d.) \(\sum_{n=0}^{\infty} n^{n} z^{n}\)

30. Consider the series \(f(z)=\sum_{n=0}^{\infty} \frac{z^{n}}{n !}\) in the region \(\{z \in \mathbb{C}: 1<|z|<2\}\). Which of the following is true?

(a.) There is a \(z\) in this region for which the series does not converge.
(b.) \(f\) is not analytic.
(c.) \(f\) takes all values in the complex plane.
(d.) \(f\) is bounded.

31. Let \(f\) and \(g\) be analytic functions on the unit disk such that \(f\left(\frac{1}{n}\right)=g\left(\frac{1}{n}\right)\) for all non-zero integers.

(a.) \(f\) and \(g\) are equal.
(b.) \(f\) and \(g\) are constant.
(c.) \(f(z)-g(z)=\sin (\pi / 2).\)
(d.) \(f(z) \neq g(z)\) if \(z\) is purely imaginary.

32. The function \(f(z)=e^{z}-2 i\)

(a.) Has no zeros.
(b.) Has only one zero.
(c.) Has a countable number of zeros.
(d.) Has an uncountable number of zeros.

33. The function \(f(z)=\tan z\)

(a.) Has poles at \(z=(2 n+1) \frac{\pi}{2}, n \in \mathbb{Z}\).
(b.) Is an entire function.
(c.) Has no zeros in \(\mathbb{C}.\)
(d.) Has a removable singularity at \(z=0.\)

34. Consider the power series \(\sum_{n=0}^{\infty} a_{n} z^{n},\) where \(a_{n}=\left\{\begin{array}{ll}\frac{1}{3^{n}} & \text { if } n \text { is even } \\ \frac{1}{5^{n}} & \text { if } n \text { is odd. }\end{array}\right.\) The radius of convergence of the series is equal to

(a.) \(r = \frac{1}{5}\)
(b.) \(r = \frac{1}{3}\)
(c.) \(r = 3\)
(d.) \(r = 5\)

35. Let \(\sum_{n=-\infty}^{\infty} a_{n} z^{n}\) be the Laurent series expansion of \(f(z)=\frac{1}{2 z^{2}-13 z+15}\) in the annulus \(\frac{3}{2}<|z|<5\). Then \(\frac{a_{1}}{a_{2}}\) is equal to

(a.) \(\frac{1}{5}\)
(b.) \(\frac{-1}{5}\)
(c.) \(-5\)
(d.) \(5\)

36. The radius of convergence of the power series \(\sum_{n=0}^{\infty} 4^{(-1)^{n} n} z^{2 n}\) is

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

37. The coefficient of \((z-\pi)^{2}\) in the Taylor series expansion of \(f(z)=\left\{\begin{array}{cc}\frac{\sin z}{z-\pi} & \text { if } z \neq \pi \\ -1 & \text { if } z=\pi\end{array}\right.\) around \(\pi\) is

(a.) \(\frac{1}{2}\)
(b.) \(-\frac{1}{2}\)
(c.) \(\frac{1}{6}\)
(d.) \(-\frac{1}{6}\)

38. Let \(\sum_{n=\infty}^{\infty} b_{n} z^{n}\) be the Laurent series expansion of the function \(\frac{1}{z \sinh z},|z|<\pi\). Then which one of the following is correct?

(a.) \(b_{-2}=1, b_{0}=-\frac{1}{6}, b_{2}=\frac{7}{360}\)
(b.) \(b_{-3}=1, b_{-1}=-\frac{1}{6}, b_{1}=\frac{7}{360}\)
(c.) \(b_{-2}=0, b_{0}=-\frac{1}{6}, b_{2}=\frac{7}{360}\)
(d.) \(b_{0}=1, b_{2}=-\frac{1}{6}, b_{4}=\frac{7}{360}\)

39. 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.

40. Let \(\sum_{n=-\infty}^{\infty} a_{n}(z+1)^{n}\) be the Laurent series expansion of \(f(z)=\sin \left(\frac{z}{z+1}\right)\). Then \(a_{-2}=\)

(a.) 1.
(b.) 0.
(c.) \(\cos (1).\)
(d.) \(\frac{-1}{2} \sin (1).\)