Free Printable Complex Integration

1.

The value of $\ \ \ \ \ \ \ \ \ \int_c^{ }4x^3dx+3y^2z^2dy+2y^3zdz$ where C is any path joining A(-1,1,0) to B(1,2,1)is

a)

0

b)

1

c)

-8

d)

8

2.

The value of $\ \ \int_c^{ }\frac{\left(3z^{2\ }+7z+1\right)}{z+1}dz$ where C is l Z l = 1/2 is

a)

$2\pi i$

b)

0

c)

$\pi i$

d)

$\frac{\pi i}{2}$

3.

$The\ value\ of\int_c^{ }\frac{\left(3z+4\right)}{z\left(2z+1\right)}dz\ \ \ \$ where C is the circle l Z l =1 is

a)

$2\pi i$

b)

$3\pi i$

c)

$4\pi i$

d)

$5\pi i$

4.

$The\ value\ of\int_c^{ }\frac{z}{\sin z}dz$ where C is l Z l = 4 is

a)

$2\pi i$

b)

0

c)

$-2\pi i$

d)

$4\pi i$

5.

$The\ value\ of\int_c^{ }z^4e\ ^{\frac{1}{z}}dz$ where C is l Z l =1 is

a)

$\frac{\pi i}{12}$

b)

$\pi i$

c)

$\frac{\pi i}{60}$

d)

$-\frac{\pi i}{60}$

6.

$The\ value\ of\ \int_c^{ }\tanh z\ dz$ where C is l Z l =3 is

a)

0

b)

$\pi i$

c)

$2\pi i$

d)

$4\pi i$

7.

$The\ poles\ of\frac{\left(z^3-1\right)}{z^3+1}$

a)

1, $\frac{1}{2}\left(-1\pm\sqrt{3}\right)$

b)

-1, $\frac{1}{2}\left(-1\pm\sqrt{3}\right)$

c)

1, $\frac{1}{2}\left(1\pm\sqrt{3}\right)$

d)

-1, $\frac{1}{2}\left(1\pm\sqrt{3}\right)$

8.

$The\ value\ of\int_c^{ }\frac{\left(4z^2+z+5\right)}{z-4}dz$ where C is $9x^2+4y^2=36$ is-----

a)

0

b)

1

c)

2

d)

3

9.

$\sum_{ }^{ }a^{n\ }z^n\ converges\ for\ \left|z\right|<R\ and\ diverges\ for\left|z\right|>R,\ then\left|z\right|=R\ is$

a)

circle of convergence

b)

radius of convergence

c)

limit of convergence

d)

boundary of convergence

10.

The sum function of the series $\sum_{n=0}^{\infty}\frac{z^n}{\lfloor n}$

a)

Exponential function

b)

Logarithmic function

c)

Sine function

d)

Cosine function

11.

$Residue\ of\ f\left(z\right)=z\cos\left(\frac{1}{z}\right)\ at\ z=0$

a)

$\frac{1}{2}$

b)

$-\frac{1}{2}$

c)

1

d)

-1

12.

$Poles\ off\left(z\right)=\frac{z^2}{z^4+1}\ in\ \left|z\right|=2$

a)

$\pm\frac{1}{2}\pm\frac{i}{2}$

b)

$\pm\frac{1}{2}\pm\frac{i}{\sqrt{2}}$

c)

$\pm\frac{1}{\sqrt{2}}\pm\frac{i}{\sqrt{2}}$

d)

$\pm\frac{1}{\sqrt{2}}\pm\frac{i}{2}$

13.

$Residue\ of\ f\left(z\right)=\frac{\left(1+e^z\right)}{z\cos z+\sin z}at\ z=0$

a)

0

b)

-1

c)

1

d)

2

14.

$The\ zeroes\ and\ \sin gularities\ off\left(z\right)=\frac{\left(z^2+1\right)}{1-z^2}$

a)

$\pm1,\pm i$

b)

$\pm2,\pm2i$

c)

$\pm i,\pm1$

d)

$\pm3,\pm3i$

15.

Taylor's series expansion of f(z)=1/z about the point z=1 is valid for

a)

$\left|z-1\right|<2$

b)

$\left|z-1\right|<1$

c)

$\left|z-1\right|<3$

d)

$\left|z+1\right|<2$

16.

$The\ \exp ansion\sum_{n=0}^{\infty}\frac{\left(-1\right)^n\left(\ z^{2n}+1\right)}{\lfloor\left(2n+1\right)},\left|z\right|<\infty\ represents$

a)

sinz

b)

cosz

c)

sinhz

d)

coshz

17.

$The\ Taylor's\ \exp ansion\ of\ f\left(z\right)=\frac{1}{\left(1+z\right)^2}\ \ \ \ \$ at z=-i is valid for

a)

$\left|z-i\right|<1+i$

b)

$\left|z+i\right|<1-i$

c)

$\left|z-i\right|<1-i$

d)

$\left|z+i\right|<1+i$

18.

$The\ Laurent's\ series\ \exp ansion\ of\ f\left(z\right)=\frac{1}{z^2-4z+3}\ about\ z=0\ is\ valid\ for$

a)

$\left|z\right|<1$

b)

$\left|z\right|<3$

c)

$1<\left|z\right|<3$

d)

None

19.

$The\ laurent's\ series\ for\ f\left(z\right)=\frac{\left(7z-2\right)}{\left(z+1\right)\left(z-2\right)}$ about z=-1 is valid for

a)

$1<\left|z+1\right|<2$

b)

$2<\left|z+1\right|<3$

c)

$1<\left|z+1\right|<3$

d)

$1<\left|z-1\right|<3$

20.

The non-isolated singularities of f(z)=cot( $\frac{\pi}{z}$ )

a)

$\pm1,\pm2,--$

b)

$\pm1,\pm\frac{1}{2},---$

c)

1,2,--

d)

0

Complex Integration

Total questions: 20

Worksheet time: 27mins

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