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

41. Consider the function \(f(x)=\frac{e^{i z}}{z\left(z^{2}+1\right)}\) the residue of \(f(z)\) at the isolated singular point in the upper half plane \(\{z=x+i y \in \mathbb{C}: y>0\}\) is

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

42. Let \(f(z)=\cos z-\frac{\sin z}{z}\) for non-zero \(z \in \mathbb{C}\) and \(f(0)=0\). Also let \(g(z)=\sinh z\) for \(z \in \mathbb{C}\), then \(\frac{g(z)}{z f(z)}\) has a pole at \(z=0\) of order

(a.) 1
(b.) 2
(c.) 3
(d.) greater than 3

43. Let \(S=\{0\} \cup\left\{\frac{1}{4 n+7}: n=1,2, \ldots ..\right\}\) then number of analytic functions which vanish only on \(S\).

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

44. It is given that \(\sum_{n=0}^{\infty} a_{n} z^{n}\) converges at \(z=3+i 4\), then the radius of convergence of the power series \(\sum_{n=0}^{\infty} a_{n} z^{n}\) is

(a.) \(\leq 5\)
(b.) \(\geq 5\)
(c.) \(<5\)
(d.) \(>5\)

45. Let \(f(z)=\frac{1}{z^{2}-3 z+2}\) then the coefficient of \(\frac{1}{z^{3}}\) in the Laurent series expansion of \(f(z)\) for \(|z|>2\) is

(a.) 0
(b.) 1
(c.) 3
(d.) 5

46. The coefficient of \(\frac{1}{z}\) in the expansion of \(\log \left(\frac{z}{z-1}\right)\) valid in \(|z|>1\) is

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

47. If \(\sin z=\sum_{n=0}^{\infty} a_{n}\left(z-\frac{\pi}{4}\right)^{n}\); then \(a_{6}\) equals

(a.) 0
(b.) \(\frac{1}{720}\)
(c.) \(\frac{1}{720 \sqrt{2}}\)
(d.) \(-\frac{1}{720 \sqrt{2}}\)

48. In the Laurent series expansion of \(f(z)=\frac{1}{z-1}-\frac{1}{z-2}\) valid in region \(|z|>2\) then the coefficient of \(\frac{1}{z^{2}}\) is

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

49. Let \(f(z)\) be an analytic function with a simple pole at \(z=1\) and a double pole at \(z=2\) with residues 1 and -2 respectively. Further if \(f(0)=0, f(3)=-\frac{3}{4}\) and \(f\) is bounded as \(z \rightarrow \infty\), then \(f(z)\) must be

(a.) \(z(z-3)-\frac{1}{4}+\frac{1}{(z-1)}-\frac{2}{(z-1)}+\frac{1}{(z-2)^{2}}\)
(b.) \(-\frac{1}{4}+\frac{1}{(z-1)}-\frac{2}{(z-2)}+\frac{1}{(z-2)^{2}}\)
(c.) \(\frac{1}{(z-1)}-\frac{2}{(z-2)}+\frac{5}{(z-2)^{2}}\)
(d.) \(\frac{15}{4}+\frac{1}{z-1}-\frac{2}{z-2}+\frac{7}{(z-2)^{2}}\)

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

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

51. Let \(f(z)=u(x, y)+i v(x, y)\) be an entire function having Taylor series expansion as \(\sum_{n=0}^{\infty} a_{n} z^{n}\). If \(f(x)=u(x, 0)\) and \(f(i y)=i v(0, y)\) then

(a.) \(a_{2 n}=0 \forall n\)
(b.) \(a_{0}=a_{1}=a_{2}=a_{3}=0 . a_{4} \neq 0\)
(c.) \(a_{2 n+1}=0\) for all \(n\)
(d.) \(a \neq 0\) but \(a_{2}=0\)

52. The radius of convergence of the power series of the function \(f(z)=\frac{1}{1-z}\) about \(z=\frac{1}{4}\) is

(a.) 1
(b.) \(1 / 4\)
(c.) \(3 / 4\)
(d.) 0

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

(a.) Removable singularity
(b.) simple pole
(c.) branch point
(d.) Essential singularity

54. The series \(\sum_{n=1}^{\infty} \frac{z^{n}}{n \sqrt{n+1}},|z \leq 1|\) is

(a.) uniformly but not absolutely convergent
(b.) uniformly and absolutely convergent
(c.) absolutely convergent but not uniformly convergent
(d.) convergent but not uniformly convergent

55. For the function \(f(z)=\frac{1-e^{-z}}{z}\), the point \(z=0\) is

(a.) An essential singularity
(b.) A pole of order zero
(c.) A pole of order one
(d.) A removable singularity.