Practice Questions for CSIR NET Complex Analysis : Analytic Functions III

21. Let \(u(x, y)=2 x(1-y)\) for all real \(x\) and \(y\). Then a function \(v(x, y)\), so that \(f(z)=u(x, y)+i v(x, y)\) 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 \(f: \mathbb{C} \rightarrow \mathbb{C}\) by \(f(z)=\left\{\begin{array}{cc}0 & \text { if } \operatorname{Re}(z)=0 \text { or } \operatorname{Im}(z)=0 \\ z & \text { otherwise }\end{array}\right.\) then the set of points where \(f\) is analytic is

(a.) \(\{z: \operatorname{Re}(z) \neq 0\) and \(\operatorname{Im}(z) \neq 0\}\)
(b.) \(\{z: \operatorname{Re}(z) \neq 0\}\)
(c.) \(\{z: \operatorname{Re}(z) \neq 0\) or \(\operatorname{Im}(z) \neq 0\}\)
(d.) \(\{z: \operatorname{Im}(z) \neq 0\}\)

Explanation for Question 22:
In part \(a\), \(f(z) = z\) is analytic in \(A\). In part \(b\), \(B\) is the set where \(\operatorname{Re}(z) = 0\) or \(\operatorname{Im}(z) = 0\), except \((0,0)\). Given \(z_{0} \in B\), we can find a sequence \(z_{n} \in A\) such that \(z_{n} \longrightarrow z_{0}\). 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 \(u_{xx}+u_{yy} \neq 0\). In part \(d\), \(u = e^{x^{2}-y^{2}\) results in \(u_{xx}+u_{yy} \neq 0\). 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 \(g(z)=x^{2}+y^{2}+i x y\) 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 \(f(z) = x^{2}+iy^{2}\), we see that \(u = x^{2}\), which results in \(u_{x} = 2x\), and \(v = y^{2}\), leading to \(v_{y} = 2y\). Since \(u_x \neq v_y\), \(f\) is not analytic. The same logic applies to function \(g(z)\).

26. Consider a function \(f(z)=u+i v\) 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.) \(\ln \left(x^{2}+y^{2}\right)\)
(c.) \(e^{x y}\)
(d.) \(e^{x^{2}-y^{2}}\)

Explanation for Question 26:
In part \(a\), \(u = x^{2}-y^{2}+xy\), we find \(u_{xx}+u_{yy} \neq 0\). For part \(c\), \(u = e^{xy}\) leads to \(u_{xx}+u_{yy} \neq 0\). In part \(d\), \(u\) involves \(e^{x^2-y^2}\), and again, \(u_{xx}+u_{yy} \neq 0\). 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 \(f(z)=z^{2} \bar{z}\)

(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 \(f(z) = (\frac{\bar{z}}{z})^2 = e^{-4i\theta}\) with \(z = re^{i\theta}\), the limit at zero depends on \(\theta\), indicating that \(f\) is not continuous at \(0\).

28. The function \(\sin z\) is analytic in

(a.) \(\mathbb{C} \cup\{\infty\}\) 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 \(\infty\) 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 \(f(z) = z\bar{z}\), it satisfies the CR equation only at \(0\), similar to Question 13.

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

(a.) \(-2 x y\)
(b.) \(x^{2}-y^{2}\)
(c.) \(2 x y\)
(d.) \(y^{2}-x^{2}\)

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

31. The function \(f(z)=\left\{\begin{array}{cc}\frac{(\bar{z})^{2}}{z^{2}} & \text { when } z \neq 0 \\ 0 & \text { when } z=0\end{array}\right.\)

(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 \(f(z) = (\bar{z}/z)^2 = e^{-4i\theta}\) has a limit at zero that depends on \(\theta\). Therefore, \(f\) is not continuous at \(0\).

32. The harmonic conjugate of \(u(x, y)=x^{2}-y^{2}+x y\) is

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

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 \(f: \mathbb{C} \rightarrow \mathbb{C}\) be given by

\(f(z)=\left\{\begin{array}{cc}\frac{(\bar{z})^{2}}{z} & \text { when } z \neq 0 \\ 0 & \text { when } z=0\end{array}\right.\)

(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 \(f(z) = \frac{\bar{z}^2}{z} = re^{-3i\theta}\) where \(z = re^{i\theta}\), 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 \(\left\{z \in \mathbb{C} \mid z^{60}=-1, z^{k} \neq-1\right.\) for \(0

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

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