Polar form of the Cauchy-Riemann equations is

\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)

\(\dfrac{\partial u}{\partial r} = r \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)

\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-\dfrac{1}{r} \dfrac{\partial u}{\partial\theta }\)

\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)

\(\dfrac{\partial u}{\partial r} = r \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=- \dfrac{1}{r} \dfrac{\partial u}{\partial\theta }\)

Assuming \(i = \sqrt { - 1} \) and t is a real number, \(\mathop \smallint \limits_0^{\frac{\pi}{3}} {e^{it}}dt\) is:

\(\frac{{\sqrt 3 }}{2} - i\frac{1}{2}\)

\(\frac{{\sqrt 3 }}{2} + i\frac{1}{2}\)

\(\frac{{\sqrt 3 }}{2} - i\frac{1}{2}\)

\(\frac{1}{2} + i\frac{{\sqrt 3 }}{2}\)

\(\frac{1}{2} + i\left( {1 - \frac{{\sqrt 3 }}{2}} \right)\)

An analytic function of a complex variable z = x + iy is expressed as f (z) = u(x,y) + iv (x, y) where i = √-1 . If u = xy, the expression for v should be

\(\frac{{{{\left( {x - y} \right)}^2}}}{2} + k\)

\(\frac{{{{\left( {x + y} \right)}^2}}}{2} + k\)

\(\frac{{{x^2} - {y^2}}}{2} + k\)

\(\frac{{{y^2} - {x^2}}}{2} + k\)

\(\frac{{{{\left( {x - y} \right)}^2}}}{2} + k\)

Let p(z) = z3 + (1 + j) z2 + (2 + j) z + 3, where z is a complex number.Which one of the following is true?

conjugate {p(z)} = p(conjugate {z}) for all z

The sum of the roots of p(z) = 0 is a real number

The complex roots of the equation p(z) = 0 come in conjugate pairs

Consider the integral \(\displaystyle\oint_C \dfrac{\sin (x)}{x^2 (x^2 + 4)}dx\)Where C is a counter-clockwise oriented circle defined as |x - i| = 2. The value of the integral is

\(\frac{\pi i}{2}-\dfrac{\pi}{8} \sin (2i)\)

An analytic function f (z) of complex variable z = x + iy may be written as f (z) = u (x, y) + iv (x, y). Then, u (x, y) and v (x, y) must satisfy

\(\frac{{\partial u}}{{\partial x}} = - \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = \frac{{\partial v}}{{\partial x}}\)

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = \frac{{\partial v}}{{\partial x}}\)

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\)

\(\frac{{\partial u}}{{\partial x}} = - \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = \frac{{\partial v}}{{\partial x}}\)

\(\frac{{\partial u}}{{\partial x}} = - \frac{{\partial v}}{{\partial y}}~and~\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\)

For a complex variable z = x + iy, which of the following statements is true?

Neither sin h Z nor cos h Z are entire functions.

Both sin h Z and cos h Z are entire functions

Neither sin h Z nor cos h Z are entire functions.

sin h Z is entire but cos h Z is not an entire function

sin h Z is not entire but cos h Z is an entire function.

Given \(f\left( z \right) = g\left( z \right) + h\left( z \right)\), where f, g, h are complex valued functions of a complex variable z. which one of the following statements is TRUE?

If \(f\left( z \right)\) is continuous at \({z_0}\), then it is differentiable at \({z_0}\).

If \(f\left( z \right)\) is differential at \({z_0}\), then \(g\left( z \right)\) and \(h\left( z \right)\) are also differentiable at \({z_0}\).

If \(g\left( z \right)\) and \(h\left( z \right)\) are differentiable at \({z_0}\), then \(f\left( z \right)\) is also differentiable at \({z_0}\).

If \(f\left( z \right)\) is continuous at \({z_0}\), then it is differentiable at \({z_0}\).

If \(f\left( z \right)\) is differentiable at \({z_0}\), then so are its real and imaginary parts.

Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is Not True?

The residue of \(\frac{{\rm{z}}}{{{{\rm{z}}^2} - 1}}{\rm{at\;z}} = 1{\rm{\;is\;}}1/2\)

\(\mathop \oint \limits_{\rm{c}}^{\rm{\;}} {{\rm{z}}^2}{\rm{dz}} = 0\)

\(\frac{1}{{2{\rm{\pi i}}}}\mathop \oint \limits_{\rm{c}}^{\rm{\;}} \frac{1}{{\rm{z}}}{\rm{dz}} = 1\)

\({\rm{\bar z}}\) (Complex conjugate of z) is an analytical function

Let z1 = 2 + 3i and z2 = 4 – 2i, then \(z = \frac{{{z_1}}}{{{z_2}}}\) will be:

\(\frac{1}{{10}} + \left( {\frac{4}{5}} \right)i\)

\(\frac{1}{{10}} - \left( {\frac{4}{5}} \right)i\)

If f (z) = u + iv is an analytic function, then

Both u and v are not harmonic functions

Both u and v are harmonic functions

Both u and v are not harmonic functions

If f(z) has a pole of order n at z = a, then residue of function f(z) at a is

\(\dfrac{1}{(n)!} \left\lbrace \dfrac{d^{n-1}}{dz^{n-1}}\left((z-a)^{n-1}f(z)\right)\right\rbrace_{z=a}\)

\(\dfrac{1}{(n-1)!} \left\lbrace \dfrac{d^{n-1}}{dz^{n-1}}\left((z-a)^{n-1}f(z)\right)\right\rbrace_{z=a}\)

\(\dfrac{1}{(n)!} \left\lbrace \dfrac{d^{n-1}}{dz^{n-1}}\left((z-a)^{n}f(z)\right)\right\rbrace_{z=a}\)

\(\dfrac{1}{(n-1)!} \left\lbrace \dfrac{d^{n-1}}{dz^{n-1}}\left((z-a)^{n}f(z)\right)\right\rbrace_{z=a}\)

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + i v(x, y), where i = √-1. If u(x, y) = x2 – y2, then expression for v(x, y) in terms of x, y and a general constant c would be

\(\frac{{{{\left( {x - y} \right)}^2}}}{2} + c\)

\(\frac{{{x^2} + {y^2}}}{2} + c\)

\(\frac{{{{\left( {x - y} \right)}^2}}}{2} + c\)

cos (z) can be expressed as

\(\frac{1}{2}\left( {{e}^{i~z}}-{{e}^{-iz}} \right)\)

\(\frac{1}{2}\left( {{e}^{i~z}}+{{e}^{-i~z}} \right)\)

\(\frac{1}{2}\left( {{e}^{i~z}}-{{e}^{-iz}} \right)\)

\(\frac{1}{2i}\left( {{e}^{i~z}}+{{e}^{-iz}} \right)\)

\(\frac{1}{2i}\left( {{e}^{i~z}}-{{e}^{-iz}} \right)\)

If the complex valued function f(z) = log z, then

f(z) satisfies Cauchy-Riemann equation

Given \(f\left( z \right) = \frac{{{z^2}}}{{{z^2} \;+ \;{a^2}}}\). Then

z = ia is a simple pole ia is a residue at z = ia of f(z)

z = ia is a simple pole and \(\frac{ia}{2}\) is a residue at z = ia of f(z)

z = ia is a simple pole ia is a residue at z = ia of f(z)

z = ia is a simple pole and \(-\frac{ia}{2}\) is a residue at z = ia of f(z)

1 + i is equivalent to

\(2{{\rm{e}}^{ - \frac{{{\rm{i\pi }}}}{4}}}\)

\(\sqrt 2 {{\rm{e}}^{ - \frac{{{\rm{i\pi }}}}{4}}}\)

\(\sqrt 2 {{\rm{e}}^{\frac{{{\rm{i\pi }}}}{4}}}\)

\(2{{\rm{e}}^{ - \frac{{{\rm{i\pi }}}}{4}}}\)

\(2{{\rm{e}}^{\frac{{{\rm{i\pi }}}}{4}}}\)

Consider the function \(f\left( z \right) = z + {z^*}\) where is a complex variable and denotes its complex conjugate. Which one of the following is TRUE?

\(f\left( z \right)\) is not continuous but is analytic

\(f\left( z \right)\) is both continuous and analytic

\(f\left( z \right)\) is continuous but not analytic

\(f\left( z \right)\) is not continuous but is analytic

\(f\left( z \right)\) is neither continuous nor analytic

If W = ϕ + iΨ represents the complex potential for an electric field.Given \(\Psi =x^2-y^2 + \dfrac{x}{x^2 +y^2}\), then the function ϕ is

\(2xy - \;\frac{y}{{{x^2} + {y^2}}}+c\)

\(- 2{\rm{xy}} + \frac{{\rm{y}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{c}}\)

\(2xy - \;\frac{y}{{{x^2} + {y^2}}}+c\)

\(2{\rm{xy}} - \frac{{\rm{x}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{c}}\)

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

−4xy + 2y2 − 2x2 + constant

Assume that Φ is harmonic in domain D and for x0 ∈ D, B(x0, r) ⊆ D, then the average value of Φ over the boundary of B(x0, r) equals

\(r{\rm{\Phi }}\left( {{x_0}} \right)\)

\({\rm{\Phi }}\left( {{x_0}} \right)\)

\(r{\rm{\Phi }}\left( {{x_0}} \right)\)

\(\frac{1}{{\pi r}}{\rm{\Phi }}\left( x \right)\)

\(\frac{4}{3}\pi {r^3}{\rm{\Phi }}\left( x \right)\)

If f(z) is analytic in domain D, then f(z) is

continuous but not necessarily differentiable in D

not both continuous and differentiable in D

If ϕ (x, y) and ψ(x, y) are functions with continuous second derivatives, then ϕ (x, y) + iψ(x, y) can be expressed as an analytic function of \(x + iy\left( {i = \sqrt { - 1} } \right)\), when

\(\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} = \frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} = 1\)

\(\frac{{\partial \phi }}{{\partial x}} = - \frac{{\partial \psi }}{{\partial x}};\;\frac{{\partial \phi }}{{\partial y}} = \frac{{\partial \psi }}{{\partial y}}\)

\(\frac{{\partial \phi }}{{\partial y}} = - \frac{{\partial \psi }}{{\partial x}};\;\frac{{\partial \phi }}{{\partial x}} = \frac{{\partial \psi }}{{\partial y}}\)

\(\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} = \frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} = 1\)

\(\frac{{\partial \phi }}{{\partial x}} + \frac{{\partial \phi }}{{\partial y}} = \frac{{\partial \psi }}{{\partial x}} + \frac{{\partial \psi }}{{\partial y}} = 0\)

If u solves ∇2u = 0, in D ⊆ Rn then,(Here ∂D denotes the boundary of D and D̅ = D ∪ ∂D)

\(\mathop {\max }\limits_{\bar D} u = u\left( x \right)\;\forall \;x \in D\)

\(\mathop {\max }\limits_{\bar D} u \ge \mathop {\max }\limits_D u\)

\(\mathop {\max }\limits_{\bar D} u = \mathop {\max }\limits_{\partial D} u\)

\(\mathop {\max }\limits_{\bar D} u = u\left( x \right)\;\forall \;x \in D\)

All the values of the multi-valued complex function 1i, where \(i = \sqrt { - 1} \), are

equal in real and imaginary parts.

71 + Mcqs in Variables in Php Page 1 McqOptions

1.	Polar form of the Cauchy-Riemann equations is
A.	\(\dfrac{\partial u}{\partial r} = r \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)
B.	\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-\dfrac{1}{r} \dfrac{\partial u}{\partial\theta }\)
C.	\(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)
D.	\(\dfrac{\partial u}{\partial r} = r \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=- \dfrac{1}{r} \dfrac{\partial u}{\partial\theta }\)
Answer» C. \(\dfrac{\partial u}{\partial r} = \dfrac{1}{r} \dfrac{\partial v}{\partial \theta} \ \text{and} \ \dfrac{\partial v}{\partial r}=-r \dfrac{\partial u}{\partial\theta }\)

2.	If f(z) = (x2 + ay2) + i bxy is a complex analytic function of z = x + iy, where \(i = \sqrt { - 1} \), then
A.	a = -1, b = -1
B.	a = -1, b = 2
C.	a = 1, b = 2
D.	a = 2, b = 2
Answer» C. a = 1, b = 2

4.	\(1\;+\;\frac{x^2}{2!}\;+\;\frac{x^4}{4!}\;+\;\frac{x^6}{6!}\;+\;.....\) stands for.
A.	sinh x
B.	cosh x
C.	cos x
D.	sin x
Answer» C. cos x

5.	If f(z) is an analytic function whose real part is constant then f(z) is
A.	function of z
B.	function of x only
C.	function of y only
D.	constant
Answer» E.

6.	Let (-1 - j), (3 - j), (3 + j) and (-1 + j) be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the countour integral \(\displaystyle\oint_C \dfrac{dz}{z^2 (z-4)}\) is
A.	0
B.	jπ/16
C.	jπ/2
D.	-jπ/8
Answer» E.

7.	If then the value of xx is
A.	\({e^{ - \frac{\pi }{2}}}\)
B.	x
C.	\({e^{ \frac{\pi }{2}}}\)
D.	1
Answer» B. x

8.	In the neighbourhood of z = 1, the function f(z) has a power series expansion of the form
A.	\(\frac{1}{z}\)
B.	\(\frac{{ - 1}}{{z - 2}}\)
C.	\(\frac{{z - 1}}{{z + 1}}\)
D.	\(\frac{1}{{2z - 1}}\)
Answer» B. \(\frac{{ - 1}}{{z - 2}}\)

10.	If C is a circle of radius r with center z0, in the complex z-plane and if n is a non-zero integer, then \(\mathop \oint \nolimits_C \frac{{dz}}{{{{\left( {z - {z_0}} \right)}^{n + 1}}}}\) equals
A.	2πnj
B.	0
C.	nj/2π
D.	2πn
Answer» C. nj/2π

11.	Let p(z) = z3 + (1 + j) z2 + (2 + j) z + 3, where z is a complex number.Which one of the following is true?
A.	All the roots cannot be real
B.	conjugate {p(z)} = p(conjugate {z}) for all z
C.	The sum of the roots of p(z) = 0 is a real number
D.	The complex roots of the equation p(z) = 0 come in conjugate pairs
Answer» B. conjugate {p(z)} = p(conjugate {z}) for all z

12.	For a complex number z = 1 - 4i with i = √-1, the value of \(\left\|\dfrac{z+3}{z-1}\right\|\) is
A.	0
B.	1/√2
C.	1
D.	√2
Answer» E.

13.	Consider the integral \(\displaystyle\oint_C \dfrac{\sin (x)}{x^2 (x^2 + 4)}dx\)Where C is a counter-clockwise oriented circle defined as \|x - i\| = 2. The value of the integral is
A.	\(\dfrac{\pi}{4} \sin (2i)\)
B.	\(\frac{\pi i}{2}-\dfrac{\pi}{8} \sin (2i)\)
C.	\(\dfrac{\pi}{8} \sin (2i)\)
D.	\(-\dfrac{\pi}{4} \sin (2i)\)
Answer» C. \(\dfrac{\pi}{8} \sin (2i)\)

22.	If z = a is an isolated singularity of f and \(f\left( z \right) = \mathop \sum \limits_{ - \infty }^\infty {a_n}{\left( {z - a} \right)^n}\) is its Laurent expansion in ann (a; 0, R), then z = a is a removable singularity if
A.	an = 0, n ≤ -1
B.	an ≠ 0, n ≤ -1
C.	an = 0, n ≥ -1
D.	an ≠ 0, n ≥ -1
Answer» B. an ≠ 0, n ≤ -1

26.	In the Laurent expansion of \(f\left( z \right) = \frac{1}{{\left( {z - 1} \right)\left( {z - 2} \right)}}\) valid in the region 1 < \|z\| < 2, the co-efficient of \(\frac{1}{{{z^2}}}\) is
A.	0
B.	1/2
C.	1
D.	-1
Answer» E.

27.	C is a closed path in the z-plane given by \|z\| = 3. The value of the integral \(\mathop \oint \nolimits_C^\; \left( {\frac{{{z^2} - z + 4j}}{{z + 2j}}} \right)dz\) is
A.	-4π (1 + j2)
B.	4π (3 – j2)
C.	-4π (3 + j2)
D.	4π (1 – j2)
Answer» D. 4π (1 – j2)

29.	If 1, ω, ω2 are cube roots of unity then the value of Δ = \(\begin{bmatrix} 1 &\omega & \omega^{2n} \\ \omega^2 & \omega^{2n} & 1 \\ \omega^{2n}& 1 & \omega^{n} \end{bmatrix} \)
A.	1
B.	0
C.	ω
D.	ω2
Answer» C. ω

18.	F(z) is a function of the complex variable z = x + iy given byF(z) = iz + k Re(z) + i Im(z).For what value of k will F(z) satisfy the Cauchy-Riemann equations?
A.	0
B.	1
C.	-1
D.	y
Answer» C. -1

19.	Let z1 = 2 + 3i and z2 = 4 – 2i, then \(z = \frac{{{z_1}}}{{{z_2}}}\) will be:
A.	\(\frac{2}{5} - \frac{1}{3}i\)
B.	\(\frac{1}{{10}} + \left( {\frac{4}{5}} \right)i\)
C.	\(\frac{2}{5} + \frac{1}{3}i\)
D.	\(\frac{1}{{10}} - \left( {\frac{4}{5}} \right)i\)
Answer» C. \(\frac{2}{5} + \frac{1}{3}i\)

20.	Let C represent the unit circle centered at origin in the complex plane, and complex variable, z = x + iy. The value of the contour integral \(\mathop \oint \nolimits_C^{} \frac{{\cosh 3z}}{{2z}}dz\) (where integration is taken counter clockwise) is
A.	2πi
B.	πi
C.	0
D.	2
Answer» C. 0

21.	If the imaginary part v = ex (x sin y + y cos y) is part of analytic function f(z) = u + iv, then f(z) is
A.	(1 + z) ez + c
B.	z ez + c
C.	z e2z + c
D.	(1 – z) ez + c
Answer» C. z e2z + c

23.	If f (z) = u + iv is an analytic function, then
A.	u is harmonic function
B.	v is harmonic function
C.	Both u and v are harmonic functions
D.	Both u and v are not harmonic functions
Answer» D. Both u and v are not harmonic functions

25.	An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + i v(x, y), where i = √-1. If u(x, y) = x2 – y2, then expression for v(x, y) in terms of x, y and a general constant c would be
A.	xy + c
B.	\(\frac{{{x^2} + {y^2}}}{2} + c\)
C.	2xy + c
D.	\(\frac{{{{\left( {x - y} \right)}^2}}}{2} + c\)
Answer» D. \(\frac{{{{\left( {x - y} \right)}^2}}}{2} + c\)

28.	Let \(f\left( z \right) = \frac{1}{{z + a}},\;a > 0\). The value of the integral ∮ f(z) dz over a circle C with center (-a, 0) and radius R > 0 evaluated in the anti-clock wise direction is
A.	0
B.	2πi
C.	-2πi
D.	4πi
Answer» C. -2πi

31.	Let z = x + jy where j = √-1. Then \(\overline {\cos z} =\)
A.	cos z
B.	cos z̅
C.	sin z
D.	sin z̅
Answer» C. sin z

32.	If the complex valued function f(z) = log z, then
A.	f(z) satisfies Cauchy-Riemann equation
B.	f(z) is analytic
C.	f(z) is not analytic
D.	None of the above
Answer» D. None of the above

36.	If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces to
A.	Cauchy’s series
B.	Maclaurin’s series
C.	Taylor’s series
D.	None of the above
Answer» D. None of the above

Explore topic-wise MCQs in Php.

Polar form of the Cauchy-Riemann equations is

If f(z) = (x2 + ay2) + i bxy is a complex analytic function of z = x + iy, where \(i = \sqrt { - 1} \), then

Assuming \(i = \sqrt { - 1} \) and t is a real number, \(\mathop \smallint \limits_0^{\frac{\pi}{3}} {e^{it}}dt\) is:

\(1\;+\;\frac{x^2}{2!}\;+\;\frac{x^4}{4!}\;+\;\frac{x^6}{6!}\;+\;.....\) stands for.

If f(z) is an analytic function whose real part is constant then f(z) is

Let (-1 - j), (3 - j), (3 + j) and (-1 + j) be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the countour integral \(\displaystyle\oint_C \dfrac{dz}{z^2 (z-4)}\) is

If then the value of xx is

In the neighbourhood of z = 1, the function f(z) has a power series expansion of the form

An analytic function of a complex variable z = x + iy is expressed as f (z) = u(x,y) + iv (x, y) where i = √-1 . If u = xy, the expression for v should be

If C is a circle of radius r with center z0, in the complex z-plane and if n is a non-zero integer, then \(\mathop \oint \nolimits_C \frac{{dz}}{{{{\left( {z - {z_0}} \right)}^{n + 1}}}}\) equals

Let p(z) = z3 + (1 + j) z2 + (2 + j) z + 3, where z is a complex number.Which one of the following is true?

For a complex number z = 1 - 4i with i = √-1, the value of \(\left|\dfrac{z+3}{z-1}\right|\) is

Consider the integral \(\displaystyle\oint_C \dfrac{\sin (x)}{x^2 (x^2 + 4)}dx\)Where C is a counter-clockwise oriented circle defined as |x - i| = 2. The value of the integral is

An analytic function f (z) of complex variable z = x + iy may be written as f (z) = u (x, y) + iv (x, y). Then, u (x, y) and v (x, y) must satisfy

For a complex variable z = x + iy, which of the following statements is true?

Given \(f\left( z \right) = g\left( z \right) + h\left( z \right)\), where f, g, h are complex valued functions of a complex variable z. which one of the following statements is TRUE?

Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is Not True?

F(z) is a function of the complex variable z = x + iy given byF(z) = iz + k Re(z) + i Im(z).For what value of k will F(z) satisfy the Cauchy-Riemann equations?

Let z1 = 2 + 3i and z2 = 4 – 2i, then \(z = \frac{{{z_1}}}{{{z_2}}}\) will be:

Let C represent the unit circle centered at origin in the complex plane, and complex variable, z = x + iy. The value of the contour integral \(\mathop \oint \nolimits_C^{} \frac{{\cosh 3z}}{{2z}}dz\) (where integration is taken counter clockwise) is

If the imaginary part v = ex (x sin y + y cos y) is part of analytic function f(z) = u + iv, then f(z) is

If z = a is an isolated singularity of f and \(f\left( z \right) = \mathop \sum \limits_{ - \infty }^\infty {a_n}{\left( {z - a} \right)^n}\) is its Laurent expansion in ann (a; 0, R), then z = a is a removable singularity if

If f (z) = u + iv is an analytic function, then

If f(z) has a pole of order n at z = a, then residue of function f(z) at a is

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + i v(x, y), where i = √-1. If u(x, y) = x2 – y2, then expression for v(x, y) in terms of x, y and a general constant c would be

In the Laurent expansion of \(f\left( z \right) = \frac{1}{{\left( {z - 1} \right)\left( {z - 2} \right)}}\) valid in the region 1 < |z| < 2, the co-efficient of \(\frac{1}{{{z^2}}}\) is

C is a closed path in the z-plane given by |z| = 3. The value of the integral \(\mathop \oint \nolimits_C^\; \left( {\frac{{{z^2} - z + 4j}}{{z + 2j}}} \right)dz\) is

Let \(f\left( z \right) = \frac{1}{{z + a}},\;a > 0\). The value of the integral ∮ f(z) dz over a circle C with center (-a, 0) and radius R > 0 evaluated in the anti-clock wise direction is

If 1, ω, ω2 are cube roots of unity then the value of Δ = \(\begin{bmatrix} 1 &\omega & \omega^{2n} \\ \omega^2 & \omega^{2n} & 1 \\ \omega^{2n}& 1 & \omega^{n} \end{bmatrix} \)

cos (z) can be expressed as

Let z = x + jy where j = √-1. Then \(\overline {\cos z} =\)

If the complex valued function f(z) = log z, then

Given \(f\left( z \right) = \frac{{{z^2}}}{{{z^2} \;+ \;{a^2}}}\). Then

1 + i is equivalent to

Consider the function \(f\left( z \right) = z + {z^*}\) where is a complex variable and denotes its complex conjugate. Which one of the following is TRUE?

If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces to

If C is a circle |z| = 4 and \(f\left( z \right) = \frac{{{z^2}}}{{{{\left( {{z^2} - 3z + 2} \right)}^2}}}\), then \(\oint f\left( z \right)dz\) is

If W = ϕ + iΨ represents the complex potential for an electric field.Given \(\Psi =x^2-y^2 + \dfrac{x}{x^2 +y^2}\), then the function ϕ is

If C is any simple closed curve enclosing the point z = z0, then the value of \( \mathop \oint \limits_C (z-z_0)~dz\)

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

Assume that Φ is harmonic in domain D and for x0 ∈ D, B(x0, r) ⊆ D, then the average value of Φ over the boundary of B(x0, r) equals

If u = x2 – y2, then the conjugate harmonic function is

Find the real values of x, y, so that – 3 + j x2y and (x2 + y) + j 4 may represent complex conjugate numbers.

If f(z) is analytic in domain D, then f(z) is

If ϕ (x, y) and ψ(x, y) are functions with continuous second derivatives, then ϕ (x, y) + iψ(x, y) can be expressed as an analytic function of \(x + iy\left( {i = \sqrt { - 1} } \right)\), when

Consider the following complex function:\(f\left( z \right) = \frac{9}{{\left( {z - 1} \right){{\left( {z + 2} \right)}^2}}}\)Which of the following is one of the residues of the above function?

If u solves ∇2u = 0, in D ⊆ Rn then,(Here ∂D denotes the boundary of D and D̅ = D ∪ ∂D)

If (x1 + j y1) = (x2 + j y2), then which of the following is true.Here \(j = \sqrt { - 1} ,x\;\& \;y\) are real numbers.

All the values of the multi-valued complex function 1i, where \(i = \sqrt { - 1} \), are

If the imaginary part of \(\frac{{2z + 1}}{{iz + 1}}\) is – 2, then the locus of the point z in the complex plane is

38.	If W = ϕ + iΨ represents the complex potential for an electric field.Given \(\Psi =x^2-y^2 + \dfrac{x}{x^2 +y^2}\), then the function ϕ is
A.	\(-2xy+ \dfrac{x}{x^2+y^2}+C\)
B.	\(- 2{\rm{xy}} + \frac{{\rm{y}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{c}}\)
C.	\(2xy - \;\frac{y}{{{x^2} + {y^2}}}+c\)
D.	\(2{\rm{xy}} - \frac{{\rm{x}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{c}}\)
Answer» C. \(2xy - \;\frac{y}{{{x^2} + {y^2}}}+c\)

40.	A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is
A.	4xy − 2x2 + 2y2 + constant
B.	4y2 − 4xy + constant
C.	2x2 − 2y2 + xy + constant
D.	−4xy + 2y2 − 2x2 + constant
Answer» B. 4y2 − 4xy + constant

42.	If u = x2 – y2, then the conjugate harmonic function is
A.	2xy
B.	x2 + y2
C.	y2 – x2
D.	-x2 – y2
Answer» B. x2 + y2

43.	Find the real values of x, y, so that – 3 + j x2y and (x2 + y) + j 4 may represent complex conjugate numbers.
A.	x = 1, y = 4
B.	x = 2, y = 1
C.	x = 2j, y = 1
D.	x = ±1, y = -4
Answer» E.

44.	If f(z) is analytic in domain D, then f(z) is
A.	continuous in D
B.	continuous but not necessarily differentiable in D
C.	not both continuous and differentiable in D
D.	differentiable in D
Answer» E.

46.	Consider the following complex function:\(f\left( z \right) = \frac{9}{{\left( {z - 1} \right){{\left( {z + 2} \right)}^2}}}\)Which of the following is one of the residues of the above function?
A.	−1
B.	9/16
C.	2
D.	9
Answer» B. 9/16

48.	If (x1 + j y1) = (x2 + j y2), then which of the following is true.Here \(j = \sqrt { - 1} ,x\;\& \;y\) are real numbers.
A.	x1 = y2
B.	x2 = y2
C.	x1 – j y1 = x2 – j y2
D.	x1 – j y1 = y2 + j x2
Answer» D. x1 – j y1 = y2 + j x2

49.	All the values of the multi-valued complex function 1i, where \(i = \sqrt { - 1} \), are
A.	purely imaginary
B.	real and non-negative
C.	on the unit circle
D.	equal in real and imaginary parts.
Answer» C. on the unit circle

50.	If the imaginary part of \(\frac{{2z + 1}}{{iz + 1}}\) is – 2, then the locus of the point z in the complex plane is
A.	x + 2y – 2 = 0
B.	2x + y – 2 = 0
C.	x – 2y – 2 = 0
D.	x + 2y + 2 = 0
Answer» B. 2x + y – 2 = 0