Download PTU B.Sc CS-IT 2020 March 6th Sem 72781 Real Analysis Question Paper

Download PTU (I.K. Gujral Punjab Technical University Jalandhar (IKGPTU) B-Sc CSE-IT (Bachelor of Science in Computer Science) 2020 March 6th Sem 72781 Real Analysis Previous Question Paper

1 | M- 72781 (S3)-1978

Roll No. Total No. of Pages : 02
Total No. of Questions : 07
B.Sc. (Computer Science) (2013 & Onwards) (Sem.?6)
REAL ANALYSIS
Subject Code : BCS-601
M.Code : 72781

Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION-B contains SIX questions carrying TEN marks each and students have
to attempt any FOUR questions.

SECTION?A
1. Write briefly :
a) State Weierstrass M-test for uniform convergence of sequence of functions.
b) Prove that f (z) = z is not analytic anywhere.
c) Show that f (z) = 1/z is not uniformly continuous in the region | z | < 1.
d) Show that cross ratio is invariant under bilinear transformation.
e) Determine the angle of rotation at z = (1 + i)/2 under the mapping w = z
2
.
f) Find the radius of convergence of the series
1
!
n
n
n x
?
?
?
.
g) Determine values of a, b such that z = ax
3
+ by
3
is harmonic function.
h) Prove that ?a
n
n
?x
is uniformly convergent on [0,1} if ?a
n
converges uniformly in
[0, 1].
i) Discuss the convergence and uniform convergence of sequence {e
?nx
}.
j) Determine the linear functional transformation that maps z
1
= 0, z
2
= 0, z
3
= 1 onto
w
1
= ? 1, w
2
= ? 1, w
3
= 1, respectively.

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1 | M- 72781 (S3)-1978

Roll No. Total No. of Pages : 02
Total No. of Questions : 07
B.Sc. (Computer Science) (2013 & Onwards) (Sem.?6)
REAL ANALYSIS
Subject Code : BCS-601
M.Code : 72781

Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION-B contains SIX questions carrying TEN marks each and students have
to attempt any FOUR questions.

SECTION?A
1. Write briefly :
a) State Weierstrass M-test for uniform convergence of sequence of functions.
b) Prove that f (z) = z is not analytic anywhere.
c) Show that f (z) = 1/z is not uniformly continuous in the region | z | < 1.
d) Show that cross ratio is invariant under bilinear transformation.
e) Determine the angle of rotation at z = (1 + i)/2 under the mapping w = z
2
.
f) Find the radius of convergence of the series
1
!
n
n
n x
?
?
?
.
g) Determine values of a, b such that z = ax
3
+ by
3
is harmonic function.
h) Prove that ?a
n
n
?x
is uniformly convergent on [0,1} if ?a
n
converges uniformly in
[0, 1].
i) Discuss the convergence and uniform convergence of sequence {e
?nx
}.
j) Determine the linear functional transformation that maps z
1
= 0, z
2
= 0, z
3
= 1 onto
w
1
= ? 1, w
2
= ? 1, w
3
= 1, respectively.

2 | M- 72781 (S3)-1978

SECTION-B
2. State and prove Cauchy?s General Principle of uniform convergence.
3. Show that the sequence {f
n
}, where
2
( )
1
n
x
f x
nx
?
?
converges uniformly on R.
4. Examine the convergence of
1
1
0
log
n
x x dx
?
?

5. a) Prove that necessary condition for f (z) = u + iv, z = x + iy, to be analytic in a domain
D are that u
x
= v
y
and u
y
= ? v
x
.
b) If u = e
x
(x cos y ? y sin y), find the analytic function u + i v.
6. If f (z) is an analytic function of z in a region D of the plane and f ?(z) ? 0 inside D, show
that the maping w = f (z) is conformal at the points of D.
7. Find Fourier expansion of
, 0
( )
, 0
x x
f x
x x
? ? ? ? ? ? ?
?
?
? ? ? ? ?
?







NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.

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