Complex integration - Complex Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2008

Let \(S\) be the positively oriented circle given by \(|z - 3i| = 2\). Then the value of \(\int_S \frac{dz}{z^2 + 4}\) is

A

\(-\frac{\pi}{2}\)

B

\(\frac{\pi}{2}\)

C

\(-\frac{i\pi}{2}\)

D

\(\frac{i\pi}{2}\)

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2

2009 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2009

Let f(z) = Σ_n=0^∞ zⁿ for z ∈ ℂ. If C : |z – i| = 2 then ∮_C f(z) dz / (z – i)⁵ =

A

2πi(1 + 15i)

B

2πi(1 – 15i)

C

4πi(1 + 15i)

D

2πi

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3

2009 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2009

Let \(u(x, y)\) be the real part of an entire function \(f(z) = u(x, y) + iv(x, y)\) for \(z = x + iy \in \mathbb{C}\). If \(C\) is the positively oriented boundary of a rectangular region \(R\) in \(\mathbb{R}^2\), then \(\oint_C \left( \frac{\partial u}{\partial y} dx - \frac{\partial u}{\partial x} dy \right) =\)

A

1

B

0

C

\(2\pi\)

D

\(\pi\)

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4

2009 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2009

The Cauchy Principal Value of the integral \( \int_{-\infty}^{\infty} \frac{\sin x}{x(x^{2}+1)} d x \) is

A

\(-2 \pi(1+2 e^{-1}) \)

B

\(\pi(1-e^{-1}) \)

C

\(2 \pi(1+e) \)

D

\(-\pi(1+e^{-1}) \)

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5

2010 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2010

Let \( I = \int_{|z| = 3} \frac{f(z)}{(z - 1)(z - 2)} dz \), where \( f(z) = \sin \frac{\pi z}{2} + \cos \frac{\pi z}{2} \) and \( C \) is the curve \( |z| = 3 \) oriented anti-clockwise. Then the value of \( I \) is

A

\(4\pi i \)

B

0

C

\(-2\pi i \)

D

\(-4\pi i \)

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6

2010 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2010

Let \( \sum_{n = -\infty}^{\infty} b_n z^n \) be the Laurent series expansion of the function \( \frac{1}{z \sinh z} \), \( 0 < |z| < \pi \). Then which one of the following is correct?

A

\(b_{-1} = 1, b_1 = -\frac{1}{6}, b_3 = \frac{7}{360} \)

B

\(b_{-1} = 1, b_1 = -\frac{1}{6}, b_3 = -\frac{7}{360} \)

C

\(b_{-1} = 0, b_1 = -\frac{1}{6}, b_3 = \frac{7}{360} \)

D

\(b_{-1} = 1, b_1 = -\frac{1}{6}, b_3 = \frac{7}{360} \)

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7

2011 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2011

Let \(f : \mathbb{C} \to \mathbb{C}\) be analytic except for a simple pole at \(z=0\) and let \(g : \mathbb{C} \to \mathbb{C}\) be analytic. Then, the value of \[\frac{\mathrm{Res}_{z=0} \{ f(z) g(z) \}}{\mathrm{Res}_{z=0} f(z)}\] is

A

\(g(0)\)

B

\(g'(0)\)

C

\(\lim_{z \to 0} z f(z)\)

D

\(\lim_{z \to 0} z f(z) g(z)\)

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8

2011 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2011

\( \operatorname{Res}_{z=2} f(z) \) is

A

\(-\frac{1}{8} \)

B

\(\frac{1}{8} \)

C

\(-\frac{1}{6} \)

D

\(\frac{1}{6} \)

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9

2011 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2011

The Cauchy principal value of \( \int_{-\infty}^{\infty} f(x) dx \) is

A

\(-\frac{\pi}{6} \sqrt{3} \)

B

\(-\frac{\pi}{8} \sqrt{3} \)

C

\(\pi \sqrt{3} \)

D

\(-\pi \sqrt{3} \)

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10

2012 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2012

Let \(\oint_C\left[\frac{1}{(z-2)^3}-\frac{(a-2)^2}{z}+4\right]dz=4\pi\), where the close curve \(C\) is the triangle having vertices at \(i\), \(\left(\frac{-1-i}{\sqrt{2}}\right)\) and \(\left(\frac{1-i}{\sqrt{2}}\right)\), the integral being taken in anti-clockwise direction. Then one value of \(a\) is

A

\(1+i\)

B

\(2+i\)

C

\(3+i\)

D

\(4+i\)

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11

2013 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2013

Let \(C\) be the contour \(|z| = 2\) oriented in the anti-clockwise direction. The value of the integral \(\oint_C z e^{3/z} dz\) is

A

\(3\pi i\)

B

\(5\pi i\)

C

\(7\pi i\)

D

\(9\pi i\)

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12

2014 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2014

Let Ω = {z ∈ ℂ : Im(z) > 0} and let C be a smooth curve lying in Ω with initial point -1 + 2i and final point 1 + 2i. The value of ∫_C (1+2z)/(1+z) dz is

A

4 - 1/2 ln 2 + i π/4

B

-4 + 1/2 ln 2 + i π/4

C

4 + 1/2 ln 2 - i π/4

D

4 - 1/2 ln 2 + i π/2

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13

2014 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2014

If a ∈ ℂ with |a| < 1, then the value of
((1-|a|²)/π) ∫_Γ |dz|/|z+a|²,
where Γ is the simple closed curve |z| = 1 taken with the positive orientation, is __________

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14

2015 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2015

Let \(\sum_{n=-\infty}^{\infty} a_n z^n\) be the Laurent series expansion of \(f(z) = \frac{1}{2 z^2 - 13 z + 15}\) in the annulus \(\frac{3}{2} < |z| < 5\). Then \(\frac{a_1}{a_2}\) is equal to ______

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15

2015 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2015

The value of \(\frac{i}{4 - \pi} \int_{|z|=4} \frac{dz}{z \cos(z)}\) is equal to ______

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16

2016 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2016

Let \(\gamma = \{z \in \mathbb{C} : |z| = 2\}\) be oriented in the counter-clockwise direction. Let \[ I = \frac{1}{2\pi i} \oint_\gamma z^7 \cos\left(\frac{1}{z^2}\right) dz. \] Then, the value of \(I\) is equal to __________

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17

2017 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2017

For \(n \in \mathbb{Z}\), define \(c_n = \frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{i(n-i)x} dx\), where \(i^2 = -1\). Then \(\sum_{n \in \mathbb{Z}} |c_n|^2\) equals

A

\(\cosh(\pi)\)

B

\(\sinh(\pi)\)

C

\(\cosh(2\pi)\)

D

\(\sinh(2\pi)\)

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18

2017 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2017

Let \(C\) be the simple, positively oriented circle of radius 2 centered at the origin in the complex plane. Then \(\frac{2}{\pi i}\int_C\left(z e^{1/z}+\tan\left(\frac{z}{2}\right)+\frac{1}{(z-1)(z-3)^2}\right)dz\) equals ________.

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19

2018 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2018

In the Laurent series expansion of \(f(z) = \frac{1}{z(z-1)}\) valid for \(|z-1| > 1\), the coefficient of \(\frac{1}{z-1}\) is

A

-2

B

-1

C

0

D

1

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20

2018 · Mathematics · Complex Analysis · Complex integration

Mathematics (MA) 2018

Let \(\Gamma\) be the circle given by \(z = 4e^{i\theta}\), where \(\theta\) varies from 0 to \(2\pi\). Then \[ \oint_\Gamma \frac{e^z}{z^2 - 2z} \, dz = \]

A

\(2\pi i(e^2 - 1)\)

B

\(\pi i(1 - e^2)\)

C

\(\pi i(e^2 - 1)\)

D

\(2\pi i(1 - e^2)\)

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