Practice Questions for CSIR NET Complex Analysis : Analytic Functions I

1. The complex analytic function \(f(z)\) with the imaginary part \(e^{x}(y \cos y+x \sin y)\) is

(a.) \(z e^{z+c}\)
(b.) \((z+c) e^{2}\)
(c.) \(z e^{z}\)
(d.) \(\left(z^{2}+z\right) e^{z^{2}+z}\)

Explanation for Question 1:
Let \(f(z) = \operatorname{Re}f(z) + i \operatorname{Im} f(z) = u(x, y) + i v(x, y)\). If \(f\) is analytic, it satisfies the Cauchy-Riemann equations. Here, we find \(v = e^{x}(y \cos y + x \sin y)\). By solving the equations, we get \(f(z) = z e^{z}\).

2. Let \(f\) be a real valued harmonic function on \(\mathbb{C}\), that is, \(f\) satisfies the equation \(\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0\). Define the functions \(g=\frac{\partial f}{\partial x}-i \frac{\partial f}{\partial y}\) and \(h=\frac{\partial f}{\partial x}+i \frac{\partial f}{\partial y}\) Then

(a.) \(g\) and \(h\) are both holomorphic functions.
(b.) \(g\) is holomorphic, but \(h\) need not be holomorphic.
(c.) \(h\) is holomorphic, but \(g\) need not be holomorphic.
(d.) both \(g\) and \(h\) are identically equal to the zero function.

Explanation for Question 2:
Given \(g = \frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}\) and \(h = \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y}\), we find that \(\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} = 0\). By applying the Cauchy-Riemann equations, we show that \(g\) is holomorphic while \(h\) does not satisfy the CR equation.

3. Let \(u(x, y)=x^{3}-3 x y^{2}+2 x\). For which of the following functions \(v\) is \(u+i v\) a holomorphic function on \(\mathbb{C}\)?

(a.) \(v(x, y)=y^{3}-3 x^{2} y+2 y\)
(b.) \(v(x, y)=3 x^{2} y-y^{3}+2 y\)
(c.) \(v(x, y)=x^{3}-3 x y^{2}+2 x\)
(d.) \(v(x, y)=0\)

Explanation for Question 3:
Since \(f\) is analytic, it must satisfy the Cauchy-Riemann equations. Given \(u(x, y) = x^{3} - 3xy^{2} + 2x\), to find the conjugate \(v(x, y)\), we find \(v = 3x^{2}y - y^{3} + 2y\). Thus, the correct answer is found by checking for which \(v\) the CR equation is satisfied.

4. Let \(p(z), q(z)\) be two non-zero complex polynomials. Then \(p(z) \overline{q(z)}\) is analytic if and only if

(a.) \(p(z)\) is constant
(b.) \(p(z) q(z)\) is constant
(c.) \(q(z)\) is a constant
(d.) \(\overline{p(z)} q(z)\) is a constant

Explanation for Question 4:
We have \(\overline{q(z)} = \overline{a_{0} + a_{1} z + \cdots + a_{n} z} = \overline{a_{0}} + \overline{a_{1}} \bar{z} + \cdots + \overline{a_{n}} \overline{z^{n}}\). Considering \(p(z) \overline{q(z)}\) as analytic, we find that \(\frac{\partial}{\partial \bar{z}} p(z) \overline{q(z)} = 0\). Hence, \(\overline{q(z)} = \overline{a_{0}}\) for a function to be analytic, it must be free from \(\bar{z}\).

5. If \(z_{1}\) and \(z_{2}\) are distinct complex numbers such that \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(z_{1}+z_{2}=1\), then the triangle in the complex plane with \(z_{1}\), \(z_{2}\), and -1 as vertices

(a.) Must be equilateral
(b.) Must be right-angled
(c.) Must be isosceles, but not necessarily equilateral
(d.) Must be obtuse angled

Explanation for Question 5:
Given \(\left|z_{1}\right| = \left|z_{2}\right| = 1\) and \(z_{1} + z_{2} = 1\), we find \(z_{1} = \frac{1}{2} + \frac{\sqrt{3}}{2}i\) and \(z_{2} = \frac{1}{2} - \frac{\sqrt{3}}{2}i\). Thus, \(z_{1} = \frac{1}{2} = z_{2}\) forms an equilateral triangle.

6. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be an analytic function. For \(z=x+i y\), let \(u, v: \mathbb{R}^{2} \rightarrow \mathbb{R}\) be such that \(u(x, y)=\operatorname{Re} f(z)\) and \(v(x, y)=\operatorname{Im} f(z)\). Which of the following are correct?

(a.) \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\)
(b.) \(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=0\)
(c.) \(\frac{\partial^{2} u}{\partial x \partial y}-\frac{\partial^{2} u}{\partial y \partial x}=0\)
(d.) \(\frac{\partial^{2} v}{\partial x \partial y}+\frac{\partial^{2} v}{\partial y \partial x}=0\)

Explanation for Question 6:
Since \(f\) is analytic, both \(u\) and \(v\) satisfy Laplace's equation (Harmonic). Also, the partial derivatives can be interchanged, i.e., \(u_{x y} = u_{y x}\) and \(v_{x y} = v_{y x}\). Therefore, \(a\), \(b\), and \(c\) are true, and \(d\) is false.

7. The number \(\sqrt{2} e^{i \pi}\) is

(a.) A rational number
(b.) A transcendental number
(c.) An irrational number
(d.) An imaginary number

Explanation for Question 7:
We have \(\sqrt{2} e^{i \pi} = \sqrt{2}(\cos \pi + i \sin \pi) = -\sqrt{2}\).

8. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a complex valued function of the form \(f(x, y)=u(x, y)+i v(x, y)\). Suppose that \(u(x, y)=3 x^{2} y\). Then

(a.) \(f\) cannot be holomorphic on \(\mathbb{C}\) for any choice of \(v\)
(b.) \(f\) is holomorphic on \(\mathbb{C}\) for a suitable choice of \(v\)
(c.) \(f\) is holomorphic on \(\mathbb{C}\) for all choices of \(v\)
(d.) \(u\) is not differentiable

Explanation for Question 8:
Given \(u(x, y) = 3x^{2}y\), we find \(u_{x x} = 6y\) and \(u_{y y} = 0\). Since \(u_{x x} + u_{y y} \neq 0\), \(f\) is not holomorphic for any \(v\).

9. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a complex-valued function given by

\(f(z)=u(x, y)+i v(x, y)\). Suppose that \(v(x, y)=3 x y^{2}\). Then

(a.) \(f\) cannot be holomorphic on \(\mathbb{C}\) for any choice of \(u\)
(b.) \(f\) is holomorphic on \(\mathbb{C}\) for a suitable choice of \(u\)
(c.) \(f\) is holomorphic on \(\mathbb{C}\) for all choices of \(u\)
(d.) \(v\) is not differentiable as a function of \(x\) and \(y\)

Explanation for Question 9:
Given \(v(x, y) = 3xy^{2}\), we find \(v_{x x} = 0\) and \(v_{y y} = 6yx\). Since \(v_{x x} + v_{y y} \neq 0\), the function \(f = u + iv\) cannot be holomorphic for any choice of \(u\).

10. The real part of the square of a complex number is equal to

(a.) The square of its real part
(b.) The square of its imaginary part
(c.) The sum of squares of real and imaginary part
(d.) The difference of squares of real and imaginary

Explanation for Question 10:
We have \(z^{2} = z \cdot z = (x + iy)(x + iy) = (x^{2} - y^{2}) + i(2xy)\).