Practice Questions for CSIR NET Complex Analysis : Analytic Functions III

21. Let $u(x,y)=2x(1-y)$ for all real $x$ and $y$. Then a function $v(x,y)$, so that $f(z)=u(x,y)+iv(x,y)$ is analytic, is

(a.) $x^2-(y-1{)}^2$
(b.) $(x-1{)}^2-y^2$
(c.) $(x-1{)}^2+y^2$
(d.) $x^2+(y-1{)}^2$

Explanation for Question 21:
For $u=2x(1-y)$, we find that $u_x=2(1-y)$ and $u_y=-2x$. The CR equation is $u_x=v_y$ and $u_y=-v_x$. Solving this system, we obtain $v_x=2x$ and $v_y=2-2y$, which leads to $v=x^2+2y-y^2+c$. Choosing $c=-1$, we find $v=x^2-(y-1{)}^2$.

22. Define $f:\mathbb{C}\to \mathbb{C}$ by $f(z)=\{\begin{array}{cc}0& \text{ if }\mathrm{Re}(z)=0\text{ or }\mathrm{Im}(z)=0\\ z& \text{ otherwise }\end{array}$ then the set of points where $f$ is analytic is

(a.) $\{z:\mathrm{Re}(z)\ne 0$ and $\mathrm{Im}(z)\ne 0\}$
(b.) $\{z:\mathrm{Re}(z)\ne 0\}$
(c.) $\{z:\mathrm{Re}(z)\ne 0$ or $\mathrm{Im}(z)\ne 0\}$
(d.) $\{z:\mathrm{Im}(z)\ne 0\}$

Explanation for Question 22:
In part $a$, $f(z)=z$ is analytic in $A$. In part $b$, $B$ is the set where $\mathrm{Re}(z)=0$ or $\mathrm{Im}(z)=0$, except $(0,0)$. Given $z_0\in B$, we can find a sequence $z_n\in A$ such that $z_n⟶z_0$. However, $f(z_n)=z_n\nrightarrow 0$, and $f(z_0)=0$, so $f$ is discontinuous on every point in $B$, making it not analytic. Additionally, $f$ cannot be analytic at $(0,0)$ because every neighborhood of $(0,0)$ contains a point from $B$.

23. Which of the following is not the real part of an analytic function?

(a.) $x^2-y^2$
(b.) $\frac{1}{1+x^2+y^2}$
(c.) $\cos x\cosh y$
(d.) $x+\frac{x}{x^2+y^2}$

Explanation for Question 23:
For part $a$, $u=x^2-y^2$ gives $u_{xx}=2$ and $u_{yy}=-2$, which leads to $u_{xx}+u_{yy}=0$. For part $c$, \(u = \frac{1}{1+x^{2}+y^{2}\) provides $u_{xx}+u_{yy}\ne 0$. In part $d$, \(u = e^{x^{2}-y^{2}\) results in $u_{xx}+u_{yy}\ne 0$. Finally, in part $b$, $u=\ln (x^2+y^2$ gives $u_{xx}+u_{yy}=0$, making it the only harmonic function in the given disc.

24. The principal value of $\log \left(i^{1/4}\right)$ is

(a.) $i\pi$
(b.) $\frac{i\pi }{2}$
(c.) $\frac{i\pi }{4}$
(d.) $\frac{i\pi }{8}$

Explanation for Question 24:
The given $i$ can be expressed as $i=e^{i\left(\frac{\pi }{2}\right)}$, and thus, $i^{1/4}=e^{i\frac{\pi }{8}}$. Taking the natural logarithm on both sides, we get $\log (i^{1/4})=i\frac{\pi }{8}$.

25. Consider the functions $f(z)=x^2+i{y}^2$ and $g(z)=x^2+y^2+ixy$ at $z=0$

(a.) $f$ is analytic but not $g$
(b.) $g$ is analytic but not $f$
(c.) Both $f$ and $g$ are analytic
(d.) Neither $f$ nor $g$ is analytic

Explanation for Question 25:
For $f(z)=x^2+i{y}^2$, we see that $u=x^2$, which results in $u_x=2x$, and $v=y^2$, leading to $v_y=2y$. Since $u_x\ne v_y$, $f$ is not analytic. The same logic applies to function $g(z)$.

26. Consider a function $f(z)=u+iv$ defined on $|z-i|<1$ where $u,v$ are real-valued functions of $x,y$. If $f(z)$ is analytic then $u$ equals to

(a.) $x^2+y^2$
(b.) $\ln (x^2+y^2)$
(c.) $e^{xy}$
(d.) $e^{x^2-y^2}$

Explanation for Question 26:
In part $a$, $u=x^2-y^2+xy$, we find $u_{xx}+u_{yy}\ne 0$. For part $c$, $u=e^{xy}$ leads to $u_{xx}+u_{yy}\ne 0$. In part $d$, $u$ involves $e^{x^2-y^2}$, and again, $u_{xx}+u_{yy}\ne 0$. However, part $b$, where $u=\ln (x^2+y^2)$, results in $u_{xx}+u_{yy}=0$, making it the only harmonic function in the given disc.

27. At $z=0$, the function $f(z)=z^2\overline{z}$

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

Explanation for Question 27:
The explanation for this question is similar to Question 13. Given $f(z)=(\frac{\overline{z}}{z}{)}^2=e^{-4i\theta }$ with $z=r{e}^{i\theta }$, the limit at zero depends on $\theta$, indicating that $f$ is not continuous at $0$.

28. The function $\sin z$ is analytic in

(a.) $\mathbb{C}\cup \{\mathrm{\infty }\}$ where $\mathbb{C}$ denotes the complex plane.
(b.) $\mathbb{C}$ except on the negative real axis.
(c.) $\mathbb{C}-\{0\}$
(d.) $\mathbb{C}$

Explanation for Question 28:
The function $\sin z$ is entire but has an essential singularity at $\mathrm{\infty }$ because $\sin (1/z)$ has an essential singularity at $0$.

29. The function $f(z)=|z{|}^2$ is

(a.) Differentiable everywhere
(b.) Differentiable only at the origin
(c.) Not differentiable anywhere
(d.) Differentiable only on the real axes

Explanation for Question 29:
Given $f(z)=z\overline{z}$, it satisfies the CR equation only at $0$, similar to Question 13.

30. An analytic function $f(z)$ is such that $\mathrm{Re}\{f^{\prime }(z)\}=2y$ and $f(1+i)=2$, then the imaginary part of $f(z)$ is

(a.) $-2xy$
(b.) $x^2-y^2$
(c.) $2xy$
(d.) $y^2-x^2$

Explanation for Question 30:
For $f(z)=x^2+i{y}^2$, we have $u=x^2$, resulting in $u_x=2x$, and $v=y^2$, leading to $v_y=2y$. Since $u_x\ne v_y$, $f$ is not analytic. The same logic applies to function $g(z)$.

31. The function $f(z)=\{\begin{array}{cc}\frac{(\overline{z}{)}^2}{z^2}& \text{ when }z\ne 0\\ 0& \text{ when }z=0\end{array}$

(a.) Is not continuous at $z=0$
(b.) It is differentiable but not analytic at $z=0$
(c.) Is analytic at $z=0$
(d.) Satisfies the Cauchy-Riemann equations at $z=0$

Explanation for Question 31:
Similar to Question 13, the given function $f(z)=(\overline{z}/z{)}^2=e^{-4i\theta }$ has a limit at zero that depends on $\theta$. Therefore, $f$ is not continuous at $0$.

32. The harmonic conjugate of $u(x,y)=x^2-y^2+xy$ is

(a.) $x^2-y^2-xy$
(b.) $x^2+y^2-xy$
(c.) $2xy+\frac{1}{2}(y^2-x^2)$
(d.) $\frac{1}{2}xy+2(y^2-x^2)$

Explanation for Question 32:
We are given the function: $u=x^2-y^2+xy$ which yields $u_x=2x+y$ and $v_y=2x+y$. Consequently, we find $v=2xy+y^2+c(x)$ from these equations. Next, $u_y=-2y+x$ implies $v_x=2y-x$, leading to $v=2xy-\frac{x^2}{2}+c(y)$. Combining these results, we find $v=\frac{y^2}{2}-\frac{x^2}{2}+2xy$.

33. Let $f:\mathbb{C}\to \mathbb{C}$ be given by

$f(z)=\{\begin{array}{cc}\frac{(\overline{z}{)}^2}{z}& \text{ when }z\ne 0\\ 0& \text{ when }z=0\end{array}$

(a.) Is not continuous at $z=0$
(b.) It is differentiable but not analytic at $z=0$
(c.) Is analytic at $z=0$
(d.) Satisfies the Cauchy-Riemann equations at $z=0$

Explanation for Question 33:
Given $f(z)=\frac{{\overline{z}}^2}{z}=r{e}^{-3i\theta }$ where $z=r{e}^{i\theta }$, the limit as $r$ tends to zero exists and is equal to 0, indicating that $f$ is continuous. The correct answer is only $a$.

34. How many elements does the set $\{z\in \mathbb{C}\mid z^{60}=-1,z^k\ne -1$ for \(0

(a.) 24
(b.) 30
(c.) 32
(d.) 45

Explanation for Question 34:
Given that $z^{60}=-1⟹z^{120}=1$ has exactly $\varphi (120)$ solutions in $\mathbb{C}$.