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

1. If \(f\) and \(g\) are analytic on the same region \(D\) and if \(f(z) g(z)=0\) for all \(z\) in the region \(D\) then

(a.) at least one of \(f\) and \(g\) vanishes identically on \(D\).
(b.) \(f\) and \(g\) both vanish everywhere on \(D\).
(c.) \(f\) vanishes only at finitely many points of \(D\) and \(g\) never vanishes.
(d.) \(f\) and \(g\) vanish only at finitely many points of \(D\).

2. The value of \(\frac{i}{4-\pi} \int_{|z|=4} \frac{d z}{z \cos (z)}\) is equal to

(a.) \(0\).
(b.) \(2\).
(c.) \(2 \pi i\).
(d.) \(4 \pi i\).

3. Let \(S\) be the disk \(|z|<2\) in the complex plane and let \(f: S \rightarrow \mathbb{C}\) be an analytic function such that \(f\left(1+\frac{\sqrt{2}}{n} i\right)=\frac{-2}{n^{2}}\) for each natural number \(n\), then \(f(\sqrt{2})\) is equal to

(a.) \(3-2 \sqrt{2}\).
(b.) \(3+2 \sqrt{2}\).
(c.) \(2-3 \sqrt{2}\).
(d.) \(2+3 \sqrt{2}\).

4. Let \(f\) be an analytic function defined on the open unit disc in \(\mathbb{C}\). Then \(f\) is constant if

(a.) \(f\left(\frac{1}{n}\right)=0\) for all \(n \geq 1\).
(b.) \(f(z)=0\) for all \(|z|=\frac{1}{2}\).
(c.) \(f\left(\frac{1}{n^{2}}\right)=0\) for all \(n \geq 1\).
(d.) \(f(z)=0\) for all \(z \in(-1,1)\).

5. Let \(\sum_{0}^{\infty} a_{n} z^{n}\) be a convergent power series such that \(\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=R>0\). Let \(p\) be a polynomial of degree \(d\). then the radius of convergence of the power series \(\sum_{n=0}^{\infty} p(n) a_{n} z^{n}\) equals

(a.) \(R\).
(b.) \(\frac{1}{R}\).
(c.) \(R \cdot d\).
(d.) \(R+d\).

6. Let \(p(x)\) be a polynomial of the real variable \(x\) of degree \(k \geq 1\). Consider the power series \(f(z)=\sum_{n=0}^{\infty} p(n) z^{n}\) where \(z\) is a complex variable. Then the radius of convergence of \(f(z)\) is

(a.) 0.
(b.) 1.
(c.) \(k\).
(d.) \(\infty\).

7. 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.

8. Let \(f\) be an entire function. Suppose, for each \(a \in \mathbb{R}\), there exists at least one coefficient \(c_{n}\) in \(f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^{n}\), which is zero. Then

(a.) \(f^{(n)}(0)=0\) for infinitely many \(n \geq 0\).
(b.) \(f^{(2 n)}(0)=0\) for every \(n \geq 0\).
(c) \(f^{(2 n+1)}(0)=0\) for every \(n \geq 0\).
(d.) There exists \(k \geq 0\) such that \(f^{(n)}(0)=0\) for all \(n \geq k\).

9. Let \(f\) be a holomorphic function on the unit disc \(\{|z|<1\}\) in the complex plane. Which of the following is/are necessarily true?

(a.) If for each positive integer \(n\) we have \(f\left(\frac{1}{n}\right)=\frac{1}{n^{2}}\) then \(f(z)=z^{2}\) on the unit disc.
(b.) If for each positive integer \(n\) we have \(f\left(1-\frac{1}{n}\right)=\left(1-\frac{1}{n}\right)^{2}\) then \(f(z)=z^{2}\) on the unit disc.
(c.) \(f\) cannot satisfy \(f\left(\frac{1}{n}\right)=\frac{(-\dot{-b})^{n}}{n}\) for each positive integer \(n\).
(d.) \(f\) can not satisfy \(f\left(\frac{1}{n}\right)=\frac{1}{n+1}\) for each positive integer \(n\).

10. 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.