North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. - university paper

7)
Z = one is a ………….. of F(z) = ( 1)2
1
z z -
.
(a) zero ( b) simple pole (c) double pole (d) None of these
8) The residue of F(z) = two 2 4
1
z z
z
-
+
at a pole of order two is …………
(a) one ( b) - one (c) two (d) None of these
9) The singular points of F(z) = ( 1)2
1
z z -
are…………………..
(a) 0 , 1, -1 ( b) 0 , 1, one (c) 1, -1 (d) None of these
III ques. of 4 marks
1) State and prove Cauchy’s integral formula for F(a).
2) State and prove Cauchy’s integral formula for F' (a).
3)
Evaluate by Cauchy’s integral formula ? +
+
C z
z
1
3
2 dz , where C is
i) the circle z =2
ii) the circle z = ½.
8
4) Evaluate ? - +
+
C z z
z z
3 2
sin cos
2
p two p 2
where C is the circle z =3
5)
Use Cauchy’s integral formula to evaluate ? -
+
C z z
z
3 two 2
1 dz, where C is the boundary of a square
with vertices one + i, -1 + i , -1 - i and one – i traversed counter clock wise.
6) State Cauchy’s integral formula for Fn(a) and use it to evaluate ?
= - 2
4
2
( 1) z
z
z
e dz.
7)
Evaluate ?
z =2
z
z
e dz. And hence deduce
i) ? ? p
p
? cos(sin ) 2
2
0
?ecos d = and ii) sin(sin ) 0
2
0
? cos ? ? =
p
e ? d
8) State Taylor’s series for F(z) about z = a and obtain the Taylor’s series expansion of F(z) = sinz in
powers of z.
9)
Evaluate ?
- = - one 2
( 1)2
sin
z z
pz
dz.Expand in Taylors series
10)
Expand in Taylor’s series:
2
1
z -
for z < 2.
11)
Expand in Taylor’s series about z = 0 , the functions F(z) =
1- z
1 and g(z) = coshz.
1 2) Expand in Taylor’s series about z = 0 the subsequent functions: (i) sin z ,(ii) sinh z , (iii) cos z.
13) Expand F(x) = ez in Taylor’s series expansion about z = 0. State the region of its validity.
14)
Expand sinz in powers of ( z -
4
p
).
15) Show that
............; 1.
3 five 7
tan
3 five 7
-1 z = z - z + z - z + z <
16) Expand in Taylor’s series:
F(z) =
( 1)( 2)
1
z - z -
for z <1.
17)
Expand F(z) =
2
1
z -
for z <2 in Taylor’s series.
18)
Expand F(z) =
2
1
z -
in Laurent’s series valid for z <2.
19)
Expand F(z) =
( five 4)
4
2
2
+ +
-
z z
z in powers of z for
(i) z <1 (ii) one < z <4 and (iii) z > 4.
20)
Expand
( 2)( 1)
2 5
2
2
- +
- +
z z
z z on the annulus one < z < 2.
9
21)
Prove that S8
=
+
-
=
- n o
n
z n
z z 1
1
4 two 4
1 where 0 < z <4.
1) obtain poles and residues at these poles of f(z) = ( 1)2
1
z. z -
also obtain the sum of these
residues.
2) obtain the sum of residues at poles of f(z) = z two a2
ez
+
3) obtain the residues of f(z) = ( 1)( 2)( 3)
2
z - z - z -
z at its poles.
4) obtain the residues of ( two 1)3
1
z +
at z = i.
5) calculate residues at double poles of f(z) = ( 4)
2 3
2
2
- +
+ +
( z i ) . z
z z
6) Use Cauchy’s integral formulae to evaluate ( Any 1 )
i) dz
( z )( z ) C
? +1 + 4
1
2 two , where C is the circle
2
Z = 3
ii) ? + C z .( z )
dz
3 four , Where C is the circle z = two .
iii) dz
( z )
ze
z
z
?
- = - one 2
1 3
.
iv) ? + C ( z )
dz
2 four 2
, Where C is the circle z - i = two .
7) Show that
3
8
1
2
4
2 ? -
=
+ ? dz ie
( z )
e
C
z
, where C is the circle z = 3.
8) Expand f(z) =
( z )( z )
z
2 3
2 1
+ +
-
in the regions:
i) z <2 , ii)2 < z <3 , iii) z >3
30) Expand :
3 2
1
z two - z +
for
i) 0 < z <1 , ii) one < z <2 and iii) z >2

Earning:   Approval pending.