Gauhati University 2007 B.Sc Mathematics FIRST - Question Paper
Monday, 21 January 2013 04:15Web
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(d) hyperbola
12. The value of
along the straight line from z =0 and z =1+i is
13. A simple pole of
(a) 0
14. Residue of
at z = three is
9 2
( )
z
z
f z
-
=
z = 2
+ C z i
dz
f (z)
5
( )p
a
5
( ) -p b
5
( ) i c
p
5
( )
i
d
-p
( (1 ) (1 ))
1
( )
3 3
2 2
f z x i y i
x y
+ - -
+
=
ÎR(a )
fd k{ (b) (a)}
b
a
a £ a -a
15. If
and C is the circle
is
16. Let
If z 0 and f (0) =0, then at the origin f is
(a) continuous and f(0) exists
(b) continuous and C-R equations are satisfied
(c) discontinuous and C-R equations are satisfied
(d) discontinuous and f(0) does not exist
PART-B (Subjective-type Questions)
(Marks: 48)
ans any 3 parts of every of the ques. Nos. 17, 18, 19, 20
17. (a) If f
and k is a number such that
for all x [a, b], then prove that
+ x + x + x + × × × two 3
4
1
3
1
2
1
1
2 2
1
a bn
n
n +
³
dx
x y
x y
dy dy
x y
x y
dx +
¹ -
+
- 1
0
3
1
0
1
0
1
0
( )3 ( )
dxdy
E
a - x - y2 two 2
u e (x cos y y sin y) x = -
(b) Prove that a monotonic increasing function which is bounded in closed interval
[a,b] is a function of bounded variation. State its total variation.
(c) State and prove Weierstrass M-test of uniform convergence of a series of
function.
(d) Show that the power series
Is uniformly convergent on [-1, k], 0
18. (a) If f and g are bounded measurable functions described on a measurable set E of
finite measure, then prove that
+ = +
E E E
(af bg) a f b g
(b) If f is bounded and integrable on [-, ] and if an, bn are its Fourier coefficients,
then prove that
is convergent
(c) Show that
(d) State and prove a theorem connecting a line integral along a closed contour with a
double integral over the domain bounded by that contour.
(e) calculate
Where E is the region bounded by the circle x2+y2= ax. 4×3=12
19. (a) obtain the analytical function f(x) = u + iv of which the real part is
(b) Prove that the Cauchy-Riemann equations can be written in polar form as
q ¶q
= - ¶
¶
¶
¶
= ¶
¶
¶ u
r
v
and
v
r r
u 1
( 1)( 3)
1
( )
+ +
=
z z
f z
1< z < 3
0 one two 2
4
= p
+
¥
x
dx
z = one and z = 2
(c) Show that
16
21
)
6
(
sin
3
6 i
dz
z
z
C
p
p =
-
(d) Expand
In a Laurent series valid for
(e) Show that 4×3=12
20. (a) State and prove Liouville’s theorem.
(b) Prove that all the roots of
z7-5z3+12 = 0
lie ranging from the circle
.
(c) State Rouche’s theorem and use it to show that every polynomial of degree n has
exactly n zeros.
(d) If f: G C is analytical, then prove that f preserves angles at every point z0 of G,
where f( z0) 0.
(e) obtain a bilinear transformation which maps the upper half of the z-plane into the
unit circle in the = 0 and is mapped into = -1. 4×3=12
| Earning: Approval pending. |
