SRM UNIVERSITY RAMAPURAM PART- VADAPALANI CAMPUS, CHENNAI – 600 600 026
Department of Mathematics Sub Title: ADVANCED CALCULUS AND A ND COMPLEX
ANALYSIS
Sub Code: 15MA102 UNIT V – COMPLEX COMPLEX INTEGRATION PART-A 1. A continuous curve which does d oes not have a point o f self intersection is called
(a) Simple curve (b)Multiple curve (c)Integral curve 2. Simple curve are also called (a) Multiple curve (b) Jordan curve (c) Integral curve 3. An integral curve along a simple closed curve is called a (a) Multiple Integral (b) Jordan Integral (c) Contour Integral 4. A region which is not simply connected is called ... region (a) Multiple curve (b) Jordan connected (c) Connected curve 5. If (a)
(b)
C
6. If
(d) None Ans : (b) (d) None Ans : (c)
(d) Multi-connected Ans : (d) is continuous at all points inside and on a simple closed curve C, then
is analytic and
f ( z )dz 0
(d) None Ans : (a)
f ( z )dz 0 (c) f ( z )dz 1 C
(d)
C
is analytic and
f ( z )dz 1
Ans : (a)
C
is continuous at all points in the region bounded by the simple closed curve
C 1 and C 2 , then (a)
f ( z )dz f ( z )dz C 1
(d)
C 2
(b)
f ( z )dz f ( z )dz (c) f ' ( z )dz f ' ( z )dz C 1
C 2
C 1
C 2
f ' ( z )dz f ' ( z )dz C 1
Ans : (a)
C 2
7. A point z 0 at which a function f ( z ) is not analytic is known as a .... of f ( z )
(a) Residue (b) Singularity (c) Integrals (d) None Ans : (b) a ) then z 8. If the principal part contains an infinite number of non zero terms of ( z a is known as (a) Poles (b) Isolated Isolated Singularity (c) Essential Singularity (d) Removable Singularity Singularity Ans : (c) z 3 9. The Singularity of f ( z ) are ( z 1)( z 2) (a) z 1,3
(b) z 1,0 (c) z 1,2 (d) z 2,3
Ans : (c)
10. A zero of an analytic function f ( z ) is a value of z for which
(a) f ( z ) 0 (b) f ( z ) 1 (c) f ( z ) 1 (d) f ( z ) 0
Ans : (a)
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 1
11. The poles of f ( z )
(a) 2
z 2 2
z
1 is z 1
sin sin
(b) 0 (c) 1 (d) None
12. The poles of f ( z )
z 2 1
Ans : (a)
is
1 z (a) 1 (b) -1 (c) 1 (d) 0 Ans : (c) 1 13. The poles of f ( z ) is z 2 and z 3 is order ... and ... respectively ( z 2) 3 ( z 3) 2 2
(a) 2,3 (b) 3,2 (c) 3,3 (d) 2,2 14. The pole for the function f ( z )
Ans : (b)
tan( z / 2) z (1 i) 2
is (1 i ) of order
(a) 0 (b) 2 (c) undefined (d) 0 15. The residue of f ( z ) cot z at each poles is (a) 0
(b) 1 (c) 1/2 (d) none 1 e z
16. The residue of f ( z )
(a) 0
Ans : (d)
sin sin z z cos z (b) 1 (c) 1 (d) undefined
Ans : (b) at the pole z 0 is
Ans : (b)
17. A singular point z z 0 is said to be an ... singular point of f ( z ) , if there is no other singular point in the neighbourhood of z 0
(a) Poles
(b) Isolated
(c) Essential
(d) Removable
Ans: (b)
18. A singular point z z 0 is said to be an ... singular point of f ( z ) , if lim f ( z ) exists and finite z z 0
(a) Poles
(b) Isolated
(c) Essential
(d) Removable
Ans: (d)
19. A singular point z singularity nor a z 0 is said to be an ... singular point of f ( z ) , it is neither an isolated singularity removable singularity
(a) Poles (b) Isolated (c) Essential (d) Removable 20. If f (a) 0 and f ' (a ) 0 , then z a is called a .... (a) Simple zero
(b) Simple curve
(c) Zero of order n
Ans: (c) (d) none
Ans: (a)
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 2
Part – B B
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 3
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 4
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 5
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 6
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 7
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 8
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 9
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 10
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 11
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 12
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 13
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 14
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 15
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 16
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 17
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 18
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 19
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 20
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 21
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 22
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 23
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 24
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 25
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 26
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 27
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 28
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 29
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 30
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 31
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 32
Prepared By Mr R.Manimaran,Assistant R.Manimaran,Assistant Professor,Department Professor,Department Of Mathematics, Mathematics, SRM UNIVERSITY, Vadapalani Campus -Chennai-26
Page 33
