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

56. Expansion of function \(f(z)=\frac{1}{(3-2 z)}\) in power of \((z-3)\) and the radius of convergence of the series so obtained are

(a.) \(-\frac{1}{3}\left[1-\frac{2}{3}(z-3)+\frac{4}{9}(z-3)^{2}-\ldots \ldots.\right]|z-3|<3\)
(b.) \(-\frac{1}{3}\left[1-\frac{2}{3}(z-3)+\frac{4}{9}(z-3)^{2}-\ldots ..\right]|z-3|<\frac{3}{2}\)
(c.) \(\frac{1}{3}\left[1+\frac{2}{3}(z-3)+\frac{4}{9}(z-3)^{2}+\ldots \ldots.\right]|z-3|<\frac{3}{2}\)
(d.) \(\frac{1}{3}\left[1-\frac{2}{3}(z-3)-\frac{4}{9}(z-3)^{2}-\ldots \ldots.\right]|z-3|<\frac{3}{2}\)

57. For the function \(f(z)=\frac{z-\sin z}{z^{3}}\), the point \(z=0\) is

(a.) A pole of order 3
(b.) A pole of order 2
(c.) An essential singularity
(d.) A removable singularity

58. The singularity of \(e^{\sin z}\) at \(z=\infty\) is

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

59. Let \(p(z)=a_{0}+a_{1} z+\ldots+a_{n} z^{n}\) and \(q(z)=b_{1} z+b_{2} z^{2}+\ldots+b_{n} z^{n}\) be complex polynomials. If \(a_{0}, b_{1}\) are non-zero complex numbers then the residue of \(\frac{p(z)}{q(z)}\) at 0 is equal to

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

60. The function \(\frac{1}{z(z-1)^{3}}\) has a pole of order \(p\) and residues \(r\) where

(a.) \(p=1 \quad r=1\)
(b.) \(p=3 \quad r=\frac{1}{3}\)
(c.) \(p=3 \quad r=1\)
(d.) \(p=1 \quad r=0\)

61. Residue of \(\frac{\sin z}{z^{4}}\) at its singularity is

(a.) 0
(b.) 1
(c.) \(-1 / 6\)
(d.) 2

62. The residue of \(e^{1 / z}\) at \(z=0\) is

(a.) 2
(b.) \(\infty\)
(c.) 1
(d.) -2

63. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be analytic for a simple pole at \(z=0\) and let \(g: \mathbb{C} \rightarrow \mathbb{C}\) be analytic. Then, the value of \(\frac{\operatorname{Res}_{z=0}\{f(z) g(z)\}}{\operatorname{Res}_{z=0} f(z)}\) is

(a.) \(g(0)\)
(b.) \(g^{\prime}(0)\)
(c.) \(\lim _{z \rightarrow 0} z f(z)\)
(d.) \(\lim _{z \rightarrow 0} z f(z) g(z)\)

64. Let \(f(z)=\frac{z}{8-z^{3}}, z=x+i y\) then \(\operatorname{Res}_{z=2} f(z)\) is

(a.) \(-\frac{1}{8}\)
(b.) \(\frac{1}{8}\)
(c.) \(-\frac{1}{6}\)
(d.) \(\frac{1}{6}\)

65. The sum of the residues at all the poles of \(f(z)=\frac{\cot \pi z}{(z+a)^{2}}\), where \(a\) is constant \((a \neq 0, \pm 1, \pm 2, \ldots)\) is

(a.) \(\frac{1}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{2}}-\pi \operatorname{cosec}^{2} \pi a\)
(b.) \(-\frac{1}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{2}}+\pi \operatorname{cosec}^{2} \pi a\)
(c.) \(-\frac{1}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{2}}-\pi \operatorname{cosec}^{2} \pi a
(d.) \(\frac{1}{\pi} \sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{2}}+\pi \operatorname{cosec}^{2} \pi a\)

66. \(\int_{0}^{2 \pi} \frac{d \theta}{13-5 \sin \theta}\) is equals

(a.) \(-\frac{\pi}{6}\)
(b.) \(-\frac{\pi}{12}\)
(c.) \(\frac{\pi}{12}\)
(d.) \(\frac{\pi}{6}\)

67. The residue of \(\frac{\sin z}{z^{8}}\) at \(z=0\) is

(a.) 0
(b.) \(\frac{1}{7 !}\)
(c.) \(-\frac{1}{7 !}\)
(d.) None of these

68. Let \(P(z), Q(z)\) be two complex nonconstant polynomials of degree \(m, \(n\) respectively. The number of roots of \(H(z)=P(z) Q(z)\) counted with multiplicity is equal to:

(a.) \(\min \{m, n\}\)
(b.) \(\max \{m, n\}\)
(c.) \(m+n\)
(d.) \(m-n\)

69. The residue of function \(f(z)=e^{-e^{1 / z}}\) at \(z=0\) is:

(a.) \(1+e^{-1}\)
(b.) \(e^{-1}\)
(c.) \(-e^{-1}\)
(d.) \(1-e^{-1}\)

70. Let \(D\) be the open unit disc in \(\mathbb{C}\) and \(H(D)\) be the collection of all holomorphic functions on it. Let

\(S=\left\{f \in H(D): f\left(\frac{1}{2}\right)=\frac{1}{2}, f\left(\frac{1}{4}\right), \ldots, f\left(\frac{1}{2 n}\right), \ldots\right\}\)

\(T=\left\{f \in H(D): f\left(\frac{1}{2}\right)=f\left(\frac{1}{3}\right)=\frac{1}{2}, f\left(\frac{1}{4}\right)=f\left(\frac{1}{5}\right), \ldots, f\left(\frac{1}{2 n}\right)=f\left(\frac{1}{2 n+1}\right), \ldots\right\}\)

Then

(a.) Both \(S, T\) are singleton sets.
(b.) \(S\) is a singleton set but \(T=\phi\).
(c.) \(T\) is a singleton set but \(S=\phi\).
(d.) Both \(S, T\) are empty.