North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. - university paper
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NORTH MAHARASHTRA UNIVERSITY, JALGAON
S.Y.B.Sc. Mathematics (Sem –II)
MTH – 221 . Functions of a Complex Variable.
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I ques. of 2 marks
1) State Cauchy’s integral formula for F(a).
2) State Cauchy’s integral formula for F' (a).
3) Evaluate by Cauchy’s integral formula ?
C
dz
z
z + 2
where C is the circle z = 1.
4) Evaluate ?
= - 2
3
2
( 1) z
z
z
e dz.
5) Evaluate ? - C
z
z
ze
( 1)3
dz where C is the circle z -1 = 2.
6) Evaluate by Cauchy’s integral formula ?
C
dz
z
ez
- 2
where C is the circle z - two = 2.
7) Evaluate by Cauchy’s integral formula ?
C
dz
z z
z
( two 3)
3 1
2 - -
-
where C is the circle z = 4.
8) Evaluate ?
C
dz
z
z
1
3
2 =
+
where C is the circle z = one / 2. Use Cauchy’s integral formula.
9) describe apower series.
10) State Taylor’s series for F(z) about z = a.
11) State Laurent’s series for F(z) about z = a.
12) Expand in Taylor’s series:
2
1
z -
for z < 2.
7
13) Expand in Laurent’s series: F(z) =
2
1
z -
valid for z < 2.
14) describe zero of an analytic function.
15) describe singular point of an analytic function.
16) State the kinds of singularities.
17) describe a pole of an analytic function.
II Multiple option ques. one mark every
1) A power series R = n
n
n a (z a)
0
- S8
=
converges if ………………
(a) z - a
2) If F(z) is an analytic function at z = a, then it has a power series expansion about z = a.
(a) Statement is actual ( b) Statement is false (d) None of these
3) The region of validity for Taylor’s series about z = 0 of the function ez is ………………
(a) z = 0 ( b) z < one (c) z < eight (d) z > 1
4) The region of validity of
1+ z
1 for its Taylor’s series expansion about z = 0 is ………………
(a) z <1 ( b) z > one (c) z = one (d) None of these
5) The expansion of
2
1
z -
is valid for ………………
(a) z <1 ( b) z <2 (c) z > three (d) None of these
6)
If F(z) =
z
sin z , then z = 0 is its ……………………….
(a) Removable singularity ( b) Isolated singularity
(c) Essential singularity (d) None of these
| Earning: Approval pending. |
