Practice Questions for CSIR NET Complex Analysis : Analytic Functions II

11. If \(f(z)=u(x, y)+i x y\) is analytic then

(a.) \(u(x, y)=x^{2}-y^{2}\)
(b.) \(u(x, y)=\frac{1}{2}\left(x^{2}+y^{2}\right)\)
(c.) \(u(x, y)=\frac{1}{2}\left(x^{2}-y^{2}\right)\)
(d.) \(u(x, y)=x^{2}+y^{2}\)

Explanation for Question 11:
For \(f(z) = u(x, y) + iv(x, y)\) with \(v(x, y) = xy\), using the CR equation, we find \(v_{x} = y\) and \(v_{y} = x\). This leads us to conclude that \(u = \frac{x^{2}}{2} - \frac{y^{2}}{2}\). Thus, the true option is only \(c\).

12. Let \(f(z)=\frac{z}{|z|}\) when \(z \neq 0\) and \(f(z)=0\) when \(z=0\). Then

(a.) \(f\) is discontinuous at 0
(b.) \(f\) is not analytic at 0 but it is continuous at 0
(c.) \(f\) is differentiable at 0
(d.) \(f\) is not differentiable at 0 but it is continuous at 0

Explanation for Question 12:
Considering \(f(z) = \begin{cases} \frac{z}{|z|} & \text{if } z \neq 0 \\ 0 & \text{if } z = 0 \end{cases}\), for \(z_{n} = \frac{1}{n} + 0i \rightarrow 0\), we find that \(f(z_{n}) \rightarrow 1 \neq f(0)\). Hence, \(f\) is discontinuous at \(z = 0\) as it does not have a limit at \(0\).

13. At \(z=0\), the function \(f(z)=z^{2} \bar{z}\)

(a.) Does not satisfy Cauchy-Riemann equations
(b.) Satisfies Cauchy-Riemann equations but is not differentiable
(c.) Is differentiable
(d.) Is analytic

Explanation for Question 13:
By checking the complex form of the CR equation, i.e., \(\frac{\partial f}{\partial \bar{z}} = 0\), we find that \(\frac{\partial }{\partial \bar{z}} z^{2} \bar{z} = z^{2} = 0\) only at \(z = 0\). Therefore, \(f\) satisfies the CR equation at \(0\) and is also a polynomial in \(z,\bar{z}\). This implies that \(f\) is differentiable at \(z = 0\), but the CR equation is not satisfied in any neighborhood of \(z = 0\), hence \(f\) is not analytic.

14. The function \(f: \mathbb{C} \rightarrow \mathbb{C}\) defined by \(f(z)=|z|\) is

(a.) Analytic when \(z \neq 0\)
(b.) Analytic everywhere
(c.) Analytic nowhere
(d.) Analytic in the region \(\{z:|z|>1\}\)

Explanation for Question 14:
Considering \(f(2) = |z|\), we observe that \(f\) is a real-valued function. A real-valued function is analytic if and only if it is constant. However, \(f\) is not constant, implying that \(f\) cannot be analytic anywhere.

15. \(\alpha \neq 1\) is a complex number such that \(\alpha^{5}=1\). Which of the following is true?

(a.) \(1+\alpha+\alpha^{2}+\alpha^{3}+\frac{1}{\alpha}=0\)
(b.) \(\alpha=\alpha^{2}\)
(c.) \(\alpha^{4}\) is also equal to 1
(d.) \(|\alpha|=5\)

Explanation for Question 15:
Given \(\alpha^{5} = 1\) and \(\alpha \neq 1\), we find that \(\alpha^{5} - 1 = 0\), leading to \((\alpha-1)(1+\alpha+\alpha^{2}+\alpha^{3}+\alpha^{4}) = 0\). However, it is also true that \(\alpha^{5} = 1\) implies \(\alpha^{4} = \frac{1}{\alpha}\). Further analysis reveals that \(\alpha^{5} = 1\) implies \(re^{i5\theta} = 1\) for \(\alpha = re^{i\theta}\), leading to the values of \(\theta = \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}\), or \(\frac{8\pi}{5}\). Hence, options \(b\), \(c\), and \(d\) are not true.

16. Suppose \(f=u+i v\) is defined in some neighborhood of \(z_{0}\) and that \(u, v, u_{x}, u_{y}, v_{x}, v_{y}\) are continuous and satisfy the Cauchy-Riemann equations at \(z_{0}\). Then read the following statements.

1. \(f\) is analytic at \(z_{0}\)

2. \(z_{0}\) is a regular point of \(f\)

3. \(f^{\prime}\left(z_{0}\right)\) exists

(a.) 1 and 2
(b.) 2 and 3
(c.) Only 3
(d.) 1, 2, and 3

Explanation for Question 16:
All statements are true, as it is a necessary and sufficient condition.

17. Which of the following functions is bounded as \(z\) varies over the complex plane?

(a.) \(e^{z}\)
(b.) \(\sin z\)
(c.) \(\frac{\sin z}{e^{z}}\)
(d.) None of the above

Explanation for Question 17:
Using the fact that a non-constant entire function cannot be bounded, since \(a\), \(b\), and \(c\) are all non-constant and entire, they are unbounded.

18. If \(z_{1}, z_{2}\) are two complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) then it is necessary that

(a.) \(z_{1}=z_{1}\)
(b.) \(z_{2}=0\)
(c.) \(z_{1}=\lambda z_{2}\) for some real number \(\lambda\)
(d.) \(z_{1} \cdot z_{2}=0\) or \(z_{1}=\lambda z_{2}\) for some real number \(\lambda\)

Explanation for Question 18:
Analyzing \(z_{1} \cdot z_{2} = 0\), we find that \(z_{1} = 0\) or \(z_{2} = 0\) (verify). Also, \(\left|z_{1}+z_{2}\right| = \left|z_{1}\right| + \left|z_{2}\right|\). Furthermore, if \(z_{1} = \lambda z_{2}\), then \(\left|z_{1}+z_{2}\right| = (\lambda + 1)\left|z_{2}\right| = \left|\lambda z_{2}\right| + \left|z_{2}\right| = \left|z_{1}\right| + \left|z_{2}\right|\). Hence, the strongest condition is that \(\left|z_{1}+z_{2}\right| = \left|z_{1}\right| + \left|z_{2}\right|\).

19. The values of \(i\) where \(i\) is the square root of -1 is

(a.) \(e^{\pi / 2}\)
(b.) \(e^{-\pi / 2}\)
(c.) \(e^{i \pi / 2}\)
(d.) \(e^{-i \pi / 2}\)

Explanation for Question 19:
Using the polar form of \(i\), we find \(i = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = e^{i\frac{\pi}{2}}\). By solving the equations, we find \(v = x^{2} - y^{2} + 2y - c\).

20. The function \(f(z)=|z|^{2}+i \bar{z}+1\) is differentiable at

(a.) \(i\)
(b.) 1
(c.) \(-i\)
(d.) No point in \(\mathbb{C}\)

Explanation for Question 20:
By applying the CR Equation in the complex form, \(\frac{\partial f}{\partial \bar{z}} = \frac{\partial}{\partial \bar{z}}(z \bar{z} + i \bar{z} + 1) = z + i = 0\). This implies that \(f\) can only be differentiable at \(z = -i\), since the CR equation is satisfied only at \(-i\). Using the same logic as in Question 13, \(f\) is differentiable at \(-i\).