Question Bank

Unit4

Academic Year 2024 – 2025

<3> Semester
Third Semester Department of EEE

MA3303-ProbabilityandComplexFunction

Regulations 2021
Part- A (2 Mark Questions)
* CO3 Number to be indicated
zdz K1 Remembering
1 ∫ z−2 *
Evaluate c where c is the circle |z|=1
2 K1 Remembering *
3 K1 Remembering *
4 K1 Remembering *
5 K1 Remembering *
6 K1 Remembering *
7 K1 Remembering *
8 K1 Remembering *
9 K1 Remembering *
10 K1 Remembering *
11 K1 Remembering *
12 K1 *
Remembering
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13 K1 Remembering *
14 K1 *
Remembering
15 K1 Remembering *
16 Remembering *
17 K1 Remembering *
18 K1 Remembering *
19 K1 Remembering *
20 K1 Remembering *

Part - B (13 Mark Questions)

* CO Number to be indicated[Marks allotted in subdivisions, if any to be indicted
within brackets]
2 K2 Rememberin
f ( z)=|z| is differentiable at
(a) Show that *
g
z=0 but not analytic at z=0.
1
z K2 Rememberin
Show that the function w=e is analytic
(b) *
everywhere in the complex plane. g
Prove that the real and imaginary parts of K2 Rememberin
(a) an analytic function are harmonic g *
2 function.
When the function f(z)=u+iv is analytic K2 Rememberin
(b) *
function are harmonic function g
An analytic function with constant K2 Rememberin
(a) *
modulus is constant. g
3
If f(z) is analytic, show then f(z) is K2 Rememberin
(b) *
constant if real part of f(z) is constant. g

Find the image of the circle |z−2i|=2

4 (a) K2 Rememberin *
g
and |z−1|=1 under the transformation

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 w=1/z.
Find the bilinear transformation that maps K2 Rememberin
(b) *
the points z=∞ ,i,0 onto w=0,i,∞ g
An analytic function with constant K2 Rememberin
(a) *
modules is constant. g
Find the analytic function whose real parts K2 Rememberin
5
is g
(b) *
e x ( x cos ey− y sin ey ) also find conjugate
harmonic.
Find the image of the infinite strips (i) K2 Rememberin
¼<y<1/2 g
(a) *
(ii) 0<y<1/2 under the transformation
6
w=1/z.
Find the bilinear transformation which K2 Rememberin
(b) *
maps z=0,z=1,z=∞ g
Find the bilinear transformation which K2 Rememberin
(a) *
maps the points ∞,2,−1to1,∞ and 0. g
7 i−z k2 Rememberin
ω=
i+ z ,maps the real axis z-
(b) Show that g *
plane into the circle |ω|=1
Find the bilinear transformation that maps K2 Rememberin
(a) *
the points z=1,i,0 onto w=0,i,1 g
2 2 K2 Rememberin
Verify that u(x,y)= x − y − y is
8
harmonic in the whole complex plane and g
(b) *
find a harmonic conjugate function v and

u.
9 Show that the map w=1/z maps the totality K2 *
(a)
of circles and lines as circles |z+1+i|=2
Rememberin
g
(b) If f(z) is analytic function with constant K2 Rememberin *

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 modulus, show that f(z) is constant. g

Find the image of the circle |z−2i|=2

K2 *
Rememberin
and |z−1|=1 under the transformation
(a)
g
10 w=1/z.
Find the bilinear transformation that maps K2 *
(b) the points 1+i ,-i, 4-i of the z-plane into
the points 0,2,i of the w-plane. Rememberin
g

Part - C (15 Mark Questions)

* CO Number to be indicated
[Marks allotted in subdivisions, if any to be indicted within brackets]
Derive C-R equations as necessary condition K2 Remembering
(a) *
for a function w=f(z) o be analytic.
1 If f(z)=u+iv is a regular function of Z in a K2 Remembering
(b) 2 2 ' 2 *
domain D, then ∇ |f ( z)| =4|f ( z)|
Determine the analytic function whose real K2 Remembering
(a) sin 2 x *
part is cosh 2 y−cos2 x
2 1 K2 Remembering
u= log( x 2 + y 2 )
Show that the function 2 is
(b) *
harmonic and determine its conjugate, also
find f(z).

Find the image of |z−2i|=2 under the K2 Remembering

(a) *
transformation w=1/z.
3
Find the bilinear transformation which maps K2 Remembering
(b) *
z=1 ,i=-1 respectively onto w= i, 0,-i.
4 An analytic function with constant modulus is K2 Remembering
(a) *
constant
(b) Show that the transformation w=1/z K2 Remembering *
transformation all circles and straight lines in

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 the z-plane into circles or straight lines in the
w-plane.
Find the bilinear transformation that maps the K2 Remembering
(a) points 1+i ,-i, 2-i of the z-plane into the points *
5 0,1,i of the w-plane.
2 K2 Remembering
If f ( z )=r (cos 2 θ+i sin pθ ) is analytic ,then
(b) *
find the value of p.

Note:
QB must contain special instructions to use Codes, Data Books, Charts, Tables, Drawing and Graph sheets to be
supplied or to be permitted in specific to that Unit can be given as a foot note

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