SHRI ANGALAMMAN COLLEGE OF ENGINEERING AND TECHNOLOGY
(An ISO 9001:2008 Certified Institution)
SIRUGANOOR, TIRUCHIRAPPALLI – 621 105
UNIT-I VECTOR CALCULUS
PART-A
01.Define divergence and curl of a vector F.
02.If F (x2 -y2 2xz) i (xz-xy+yz)j (z2 +x 2 )k,find grad(divF)
03.If F xyi yz j zx k , find div(curl F )
04.If
=yz i +xz j + xy k , then find
05.Find ,such that F (3x 2 y z) i (4x + y-z)j (x-y+2z)k is solenoidal
06.Prove that F (3x 2 y 4z) i (2x +5y+4z)j (4x+4y-8z)k is both solenoidal
and irrotational
07.Find a so that the vector A (ax2 y2 x) i (2xy +y)j is irrotational
is solenoidal find 2
08.If
09.If F 3x i +x j +y k,show that curl curlF 0
10.If 2 0, show that
is both solenoidal and irrotational.
11.Find r 3 where r= r and r x i + y j +z k
xyz 2 at (1,0,3) ?
12.Find the greatest rate of increase of
13.Find the unit normal vector to the surface x 2 +xy+z 2 = 4 at the point (1,-1,2)
14.Find a unit vector normal to the surface x2y +2xy =4 at the point(2,-2,3).
15.If F x 2 i xy 2 j, evaluate
F . d r from (0,0) to (1,1) along the path y=x.
C
16. If F x2 i +xy2 j , evaluate the line integral F . dr from (0,0) to (1,1) along
the path
y=x.
17. If S is any closed surface enclosing a volume V and
F x i +2 y j +3z k
prove that
F . n dS = 6 V
18.Using Green’s theorem prove that the area enclosed by a simple closed
curve C is
19. Find
1
(x dy - y dx)
2
r .ds where S is the surface of the tetrahedron whose vertices
are (0,0,0) (1,00)(0,1,0) (0,0,1)
20.If r x i + y j +z k and S is the surface of the sphere of unit radius, find
r ds
S
/2/
21.For any closed surface S, prove that
culr F . n ds
0
S
22. Use Gauss divergence theorem, prove that
r . n ds 3 V where V is the
S
volume enclosed by the surfaces S.
PART-B
01.Find the directional derivative of xy 2 +yz3 at (2,-1,1) in the direction of the
normal to the surface xlog z - y 2 +4=0 at (-1,2,1)
= 2xyz 3 i x 2 z 3 j 3x 2 yz 2 k find (x,y,z) , given that (1,-2,2) =4
02.If
03.Find f(r) if the vector f(r) r is both solenoidal and irrotational
04.Show that F (2 xy z 2 )i ( x 2 2 yz ) j ( y 2 2 zx)k is irrotational and hence
find the
scalar function such that F
05.Prove that F (2 xy z 3 )i x 2 j 3xz 2 k irotational. Find its scalar potential
and the work done by the force F in moving a particle from (1,-2,1) to
(3,2,4)
06.Prove that F (y2cos x z3 ) i (2y sin x-4)j 3xz2 k is irrotational and
find
the scalar
potential
07. Determine f(r ) so that the vector f(r ) r is both solenoidal and
irrotational.
08.Show that div (u x v ) v .curl u u .curl v
09.Find the directional derivative of = xy 2 z 3 at the point (1,1,1) along the
normal to the
surface x 2 xy + z 2 3, at the point (1,1,1)
10.Find the directional derivative of xy+yz+zx at (1,2,0) in the direction of
i + 2j +2k
11.If r
xi + y j +zk and r = r prove that r n
12.Find the directional derivative of
nr n 2 r and curl(r n r )
0
3 x 2 +2y-3z at (1.1.1) in the direction of
2i + 2j -k
13.Find the directional derivative of = 3x 2 yz +4xz 2 +xyz at (1,2,3) in the
direction of 2i + j-k.
14.Find the angle between the surfaces x 2 +y2 +z 2 =9 and z=x 2 +y2 3 at the point
(2,-1,2)
15.Find the angle between the surfaces x 2 + y2 -z 3 and x 2 + y 2 +z 2 =9
at (2,-1,2)
16.Find the area between the surfaces x 2 y 2 z 2 9 and x 2 y 2 3 z at the point
(2,-1,2).
/3/
17. Verify Gauss divergence theorem for F 4 xzi y 2 j yzk over the cube
x=0,x=1,y=0,y=1
and z=0,z=1
18.Verify Gauss’s Divergence theorem for F x2 i y2 j z2 k taken over
the
cube
bounded by the planes x=0,x=1,y=0,y=1,z=0 and z=1.
19. Use Gauss divergence theorem to evaluate f . n ds where
S
2
4xi 2y j z k and S
f
x
2
2
y
2
is the surface bounding the region
4, z=0 and z=3
F ds,if F ( x 2
20.Evaluate
yz)i ( y 2
zx) j ( z 2
xy)k where S is the surface
S
bounded by
0 x a,0 y b,0 z c
21.Verify Green’s theorem on the plane for
xy
y 2 dx x 2 dy where C is the
c
closed curve of
the region bounded by y=x and y x 2
22.State Green’s theorem in the plane. Using Green’s theorem, evaluate
(3x 2 8 y) dx (4y-6xy)dy , where C is the boundary of the region enclosed
C
by y=x 2 and x=y 2
23.Evaluate [(2xy-x 2 ) dx (x+y2 )dy] using Green’s theorem where C is
C
closed curve formed
by y=x 2 and x=y 2
24. Verify Stoke’s theorem for F yi z j xk , where s is the upper half of
x 2 y2 z2 1
sphere
25.Verify Stoke’s theorem for a vector field F (x2 y2 ) i 2xy j in the
rectangular region
bounded by the lines x a, y 0, y b
the
26.Verify Stoke’s theorem to find F . d r where F xi 2 y j 3zk and C is the
C
2
2
curve x +y =4 and z=2
27.Verify Stoke’s theorem for a vector field F (x2 -y2 ) i 2xy j in the
rectangular region
of the
z=0 plance bounded by the lines
x=0,x=2,y=0 and y=3.
28.Verify Stoke’s theorem for F xy i 2yz j zx k where S is the open
surface of the
rectangular parallelepiped formed by the planes
x=0,x=1,y=0,y=2 and z= 3 above
the
XOY Plane
29. Using Stoke’s theorem evaluate F .dr , where F (2x-y) i yz 2 j y 2zk
C
where C is the circle x
of unit radius
2
y
2
1, corresponding to the surface of the sphere
/4/
30. Verify Stoke’s theorem for F ( y z+2)i (yz+4) j - x z k over the open
surface of the cube
x=0,y=0,x=1,y=1,z=1 not included in the XOY
plane.
31.Evaluate F .n ds,if F xi 2 y j 3zk and S is the closed surface of the
S
x2
surface
y2
z2
a2
32. Prove that the area enclosed by a simple closed curve C is
1
(x dy - y dx)
2
using Green’s theorem. Hence the area of a circle of radius ‘ r’ units.
33. If F (3x 2 6 y) i 14yz j 20xz 2 k,evaluate F .dr from (0,0,0) to (1,1,1) along
x=t, y=t 2 , z=t 3
A.n ds where A= 4xz i - y 2 j + yz k and S is the surface of the
the curve
34.. Evaluate
S
cube bounded by x=0,s=z,y=0,y=1, z=0 and z=1.
35. If S is any closed surface enclosing a Volume V and if
F ax i by j cz k, Prove that
A.n ds = (a+b+c)v
S
36. Find the work done by the force F z i x j yz k where it moves a particle
along the arc of the curve r cos t i sin t j t k from t 0 t= 2 .
37. Evaluate (ex dx 2ydy- dz), by using Stoke’s theorem where C is the curve
C
x
2
y
2
4, z=2
38. Evaluate
f . n ds where f
(x+y2 )i 2x j 2yz k
where S is the surface of the
S
2x+y+2z = 6 in the first octant.
39.If A 2xy i (x 2 +2yz)j (y2 +1)k find the value of
A .dr
C
circle with
centre at the origin in the xy plane.
around the unit
/5/
UNIT-II- DIFFERENTIAL EQUATIONS
PART-A
2
dy
y
dx 2
2
02. Solve (D 2)y = 0
03. Solve (D3 1)y = 0
2
2
04. Solve (D 1) y = 0
4
3
2
05. Solve (D 2D +D )y = 0
01. Solve
11
2y1 +5y = e-xcos 2x
2
3
07. Find the particular integral of the differential equation (D 1)y = x
2
3x
08. Find the particular integral of (D -3) y = e cosx
3
-2x
09. Find the particular integral of (D 8)y = e
2
x
10. Find the particular integral of (D 2D +5)y = e sin 2x
2
11. Find the particular integral of (D 4)y = sin 2x
3
12. Find the particular integral of (D-1) y = 2 coshx
2
-2x
13. Find the particular integral of (D 4D +4) = xe
d 2 u du
14. Solve r 2
0
dr
dr
11
15. Transform the equation x y
y1 +1 = 0 in to a linear equation with constant coefficients.
d2y
z
16. If x=e , express
in terms of the derivatives of y with respect to z.
dx 2
06. Find the particular Integral of y
17. Write Euler’s Homogeneous linear differential equation. How will you convert it to a. linear
differential equation with constant coefficients?
d2 y
dy
18. Solve x
4x
2y = 0
2
dx
dx
dx
dy
19. Solve
y = 0,
+x = 0
dt
dt
2
20. Determine the part of the complementary function of
d2 y
dx 2
dy
2(1 x)y = x 3
dx
2
dy
2 d y
21. Solve x
x
0
dx 2
dx
11
1
22. Transform the equation xy + y +1 = 0 into a linear equation with constant
x2
2x(1+x)
coefficient.
2 11
1
23. Solve x y + 2xy = 0
/6/
xy11 + y1 = 0
2
dy
2 d y
25. Solve (2x+3)
2(2x+3)
12y = 0
2
dx
dx
24. Solve
PART-B
11
1
01.Solve y +2y +y = x cos x
2
5D +4)y = e-x sin 2x x 2 1
2
2x 4
03. Solve (D 4D +4)y = e x
cos2x
2
2
4
04 Solve (D 4)y = cos x x
3
2
-x 3
05. Solve (D 3D +3D-1)y = e x
2
2x
06. Solve (D 4D +13)y = e cos3x
2
2
07. Solve (D 5D +6)y = cos 2x x
4
3
2
08. Solve (D D D )y = 2cos 2x cos x
2
09. Solve (D +4)y = x sin x
d2 y
10. Solve
4 y x sin h x
dx 2
2 11
1
4
11. Solve x y -2xy +y = x
2
2
12. Solve (x D xD +1)y = x log x
02. Solve (D
2
2
13. Solve (x D
2
14.Solve (x D
18. Solve
19. Solve
20. Solve
2
4xD +2)y = x 2 +
dx
dy
+5x-2y = t ,
+2x +y= 0
dt
dt
dx
dy
+2y = 5et ,
-2x = 5et given x=-1,y=3 when t=0
dt
dt
21. Solve for y, from
22. Solve
2
1
x2
(x 2D2 -4xD +6)y = sin(log x)
(x 2D2 2xD -4)y = 32(log x)2
(x 2D2 +xD +1)y = log x sin(log x)
(x2D2 -3xD -5)y = sin(log x)
(x2D2 xD +1)y = log x
2
17. Solve
log x
xD -1)y =
x
2
15. Solve (x D
16.Solve
4xD +6)y = x(1+x)
/7/
23. Solve
(D+4) x +3y = t, (D+5)y +2x = e2 t
24. Solve
dx
dy
1
+y = e2t ,
+4x = t, given that x(0) = 2 & y(0) =
dt
dt
4
dy
dx dy
+2y = cos 2t,
-2x = sin 2t
dt
dt dt
dx
dy
26.Solve the simultaneous equation
+2y +sin t=0 ,
-2x-cos t =0
dt
dt
25. Solve
dx
dt
t
27. Solve (2D+1) x +(3D+1)y = e , (D+5)x +(D+7)y = e
2
t
2
3(3x+2)D -36]y = 3x 2 4x+1
11
1
2x
29. Solve y +7y -8y = e
by method of variation of parameters
2
2
30. Solve the equation (x D xD +1)y = x log x by the method of variation of
28. Solve [(3x+2) D
parameters.
11
1
x
31. Solve y -2y +2y = e tan x by method of variation of parameters
d2 y
32. Solve
dx 2
11
33. Solve y -y =
4y = sec 2x by variation of papameter method.
2
1+ex
by the method of variation of parameters
d2 y
34. Using the method of variation of parameters, solve
dx 2
4y
tan 2x
.
UNIT –III LAPLACE TRANSFORM
Part- A
01.Write a function for which LT does not exist. Explain why LT does not exist
-2t
2
02.Find the LT of e t
03.Find the LT of t sin at
-t
04.Find the LT of te sin t
05.If L[f(t)] = F(s) what is L e-at f (t) ?
t
te t dt
06.Find the Laplace transform of
0
1
s
F
a
a
07. If L[f(t)] = F(s), Prove that L f(at)
08.State both the initial and final value theorem of Laplace Transform
09. Find L-1
1
(s-2)3
s+1
(s +2s+2)
1
11. Find the inverse LT of
(s+2) 4
10. Find the inverse LT of
12.Find
L-1
2
s+2
(s +1)2
2
13. Find the inverse L.T. of
s
( s 2)3
14.If L{f(t)}=
1
, find lim f (t )
t
s ( s 2)
15.If L{f(t)}=
s+2
, find e t f (t )dt
2
s 4
0
PART-B
01. Find the L.T. of
1 cos t
s
, L 1 tan -1
t
a
and (iii) L 1 cot
e t sin 2t
t
02.Find the L.T. of (i) te 4t sin 3t and (ii)
t
03.Find the LT of
e-t sin t
dt
t
0
t
-3t
04. Find the L.T. of t e sin 2t and e
-t
sin 2t
dt
t
0
1
s
k
/2/
0 t
a sin at
0
05.Find the L.T. of f (t )
E
E
06.Find the L.T. of f (t )
sin t
08.Find the LT of f(t) =
0 t
a
a
t
2
for 0<t<
07. Find LT of f(t) =
0
2
a
and f(t+ 2
t< 2
for
t
2
t
for 0<t<
) = f(t)
and f(t+2 ) f(t)
2 -t for <t<2
09. Find the L.T. of the periodic function defined by the triangular
t
for 0<t<a
a
and f(t+2a) f(t)
wave f(t) =
(2a-t)
for a<t<2a
a
10.Verify Final value theorem for the function f (t ) 1 e 2t
11.Find the inverse LT of
12.Find the inverse L.T. of
13.Find L-1
(s 2
s e -s
( s 3)5
1
s( s
2
a2 )
s2
a 2 )( s 2 b2 )
s
(s +a 2 )2
s-5
15.Find the inverse LT of log 2
s 9
14.Find the inverse LT of
2
16.Find the inverse L.T. of
2
and cot -1
s 1
5s-13
and 2
s 6s 25
1
s (s
2
2
a2 )
using convolution theorem
17.Using Convolution theorem, find L-1
s
( s a 2 )2
18.Using Convolution theorem find L-1
1
( s 1)( s 2)
2
19.Stat convolution theorem, using this, find L-1
1
(s +a 2 )2
2
/3/
20.Using convoloution, solve the initial value problem
y''+9y sin 3t given y(0) =0, y'(0) = 0
t
21. Solve, using L.T., y+ ydt 1 et
0
t
22. Using L.T., find the solution of y'+3y+2 y dt t given y(0) =0
0
2
d y
dt 2
d2 y
24. Solve using L.T., 2
dt
d2 y
25. Solve using L.T. 2
dt
d2 y
26..Solve, using LT 2
dt
d2 y
27. Solve, using LT 2
dt
23. .Using L.T. solve,
dy
+2y 4 , given y(0) =2, y'(0) = 3.
dt
dy
3 +2y e3t , given y(0) =1, y'(0) = -1.
dt
dy
3 +2y e3t , given y(0) =1, y'(0) = 0.
dt
dy
6
8 y 2 given y(0) = 1, y'(0) = 1
dt
dy
2y 3cos 3t 11 sin 3t given y(0) = 0, y'(0) = 6
dt
3
UNIT- IV ANALYTIC FUNCTIONS
PART –A
01.State Cauchy-Riemann equation
02.Prove that z 3 is analytic
03.Show that an analytic function with constant real part is constant.
04.If u+iv is analytic, show that v-iu and –v+iu are also analytic
05.Give an example such that u and v are harmonic but u+iv is not analytic
06.Prove that zz is not analytic
z2 4
07.Find, where the function f(z) = 2
ceases to be analytic
z +1
08.State any two properties of analytic function
09.If f(z) = u+iv be an analytic function, prove that u satisfied the
Laplace
equation
10.Verify the function (x,y) = e x sin y is harmonic or not
11.Verify whether e x cos y is harmonic
12.Show that u= log (x 2 +y 2 ) is harmonic
13.Verify the function u(x,y) = log x 2 +y2 is harmonic or not?
14.Find the value of ‘m’ so that the function u= 2x + x 2 my 2 is harmonic
15.Find the constant a so that u(x,y) = ax 2 y 2 xy is harmonic.
16. State the orthogonal property of analytic function
z-1
z+1
17.Find the fixed points of the transformation w =
18.Find the invariant point of the bilinear transformation w =
1+z
1-z
6z-9
z
19.Find the fixed point of the transformation w =
20.Determine the critical points of w = (z- )(z- )
2
21.Find the critical points of the mapping w = z
22.Find the invariant point of the transformation w=
23.Find the critical points of the mapping w = z+
z+1
2z+1
1
z
24.Define conformal mapping
25.Find the image of the region x>c where c>0 under the
transformation w =
1
z
26.Find the image of z-2i
2 under themapping w =
1
z
27.By the transformation w = iz show that the half plane x>0 maps in to the
half plane w>1.
28.Find the image of 2x +y -3 = 0 under the transformation w = z+2i
29.Obtain the image of z 1 under themapping w = 2z
30.Find the map of the circle z
3 under themapping w = 2z
PART B
01.Show that sinh z is an analytic function and find its derivative
02.Show that the function w = z n where n is a positive integer, is analytic
dw
dz
everywhere in the complex plane and find its derivative
03.Show that an analytic function with constant real part is constant.
04.Prove that an analytic function in a region with constant modulus is
constant.
05.Prove that an analytic function with constant imaginary part is constant..
06.Show that the function f(z) = z z is differentiable but not analytic at
z=0
x
07.Prove that the function f(z) = e (cos y i sin y) is nowhere differentiable.
2
08.If f(z) is a regular function of z, prove that
2
09.If w=f(z) is analytic then prove that
x
2
x
2
2
y
2
f(z)
2
2
y
2
log f ' (z)
0
4 f ' (z)
2
3
2
2
2
10.Show that u= x -3xy + 3x -3y +1 is harmonic and its harmonic
conjugate
function.
11.Show that the function u(x,y) = 3x 2 y+2x 2 -y3 -2y2 is harmonic. Find also the
conjugate
harmonic function
12.Show that the function u(x,y) = x 3 3xy2 +x 2 -y2 +2xy is harmonic and find the
corresponding analytic function f(z) = u+iv.
13.Show that u(x,y) = ex (x cos y - y sin y) is harmonic. Also find the analytic
function f(z) = u+iv
14.If u= x 2 - y2
but u+iv
and v=
is
y
Prove that u and v are harmonic functions
x 2 +y2
not a regular function of z.
x
x +y 2
16.Find the analytic function whose real part is e2x (x cos 2y + y sin 2y)
15.Construct the analytic function f(z) = u+iv, if u+v =
2
17.Find the analytic function whose imaginary part is ex (x sin y + y cos y)
2
2
18.If f(z) = u+iv is analytic and u-v= (x-y)(x +4xy + y ) find f(z) in terms
of z.
x
*
x +y2
sin 2x
20.Find the analytic function f(z) = u+iv , given that u =
cosh 2y+cos 2x
2
2
19.Find the analytic function f(z) = u+iv give v= x - y +
2
21.If u= log (x 2 +y 2 ) find v and f(z) such that f(z) = u+iv is analytic
22.Use Milne Thompson method to find the harmonic conjugate u of
v= e-x [2xy cos y+(y 2 x 2 ) sin y] where u+iv is the analytic function.
23.If u+v =
sin 2x
and f(z) = u+iv is analytic, find f(z)
cosh 2y - cos 2x
24. Construct the analytic function f(z) = u+iv given that
2u +3v = e x (cos y
i sin y)
25.Find the bilinear map which maps the points z=1,-i,-1 onto the points
w = i,0,-i respectively
26.Find the bilinear transformation which maps the points z=1,i,-1 into the
points w=i,0,-i. Hence find the image of z 1
27.Find the bilinear transformation which maps the points z= 0,,-1, i onto
z 1
w= i,0, .Also find the image of the unit circle
28.Find the bilinear transformation which maps z= 1,0,,-1into w= 0,-1,
respectively. What are the invariant points of the transformation.
29.Find the bilinear mapping which maps -1,0,1 of the z-plane onto -1,-i,1 of
the w- plane.
30. Find the bilinear map which maps the points z=1,i,-1 onto the points
w = i,0,-i.
31.Find the bilinear transformation which maps 1,i and -1 of the z-plane into
0,1 and of the w-plane. Show that the transformation maps
the
interior of the unit circle of
the z-plane onto the upper half of the
w- plane.
32.Find the bilinear transformation which maps the points z= 0,1, into
the
points w= i,1 and –i
33.Find the bilinear transformation that maps the points 1+i,-i , 2-i at the z
plane into the
points 0,1,i of the w plane
34.Show that the transformation w =
the
z-plane onto
1
maps circles and straight lines of
z
circles or straight lines of the w-plane.
35.Find the image of z-3i 3 under the mapping w =
1
z
36.Find the image of the circle z-1 1 under the mapping w =
Show the result graphically
1
z
(rough sketch)
37.Find the image of x+y = 1 under the transformation w = z 2
38.Find the image of the circle z-3i 3 under the mapping w =
1
z
39.Find the image of the region bounded by the lines x= 0, y=0 and x+y = 1 in
the z-plane by
the mapping w = z ei /4
40.Determine the region of the w-plane into which the first quadrant of z-plane
mapped by the transformation w=z 2
UNIT- V COMPLEX INTEGRATION
PART –A
2
1 z +5
dz where C is z 4 using Cauchy’s integral formula
2 i C z-3
dz
where C is the circle z-2 1
02.Evaluate
z-2
C
01.Evaluate
4
03. Evaluate
z-3 dz where C is the circle z-3
C
04.Evaluate
z dz
where C is the circle z
(z-3) 4
C
1
2
05. Evaluate
dz
where C is the circle z
2z-3
C
1
/14/
2
06. Evaluate
z +1
dz where C is a circle of unit radius and centre at z=1
z 2 -1
C
z 2 2 z dz where C is the circle z
07. Evaluate
1
C
08. Evaluate
2
dz where c is |z-1|=2
( z 1)( z 2)
C
dz
where z
sin z
C
09. Evaluate
1
1 i
( x y ix 2 )dz . Along the line joining z=0 and z=1+i
10. Evaluate
0
(z2 +2z+1) dz where C is the circle z
11. Evaluate
1
C
12.Obtain the expansion of log(1+z) when
z
1
1
at z=1 in a Taylor’s series
(z-2)
1
14.Expand
as Laurent’s series about z=0 in the annulus 0 z 1
z(z-1)
1
15.Find the Laurents expansion of 2
in z 1 .
z (1-z)
13. Expand
16.Expand sin z as a Taylor’s series at z
17.Find the poles of f(z) =
4
1
sin
1
z-a
18.Find the residue of f(z) = cot z at its pole.
19.Find the singular points of f(z) =
1
z sin z
20.Define essential singularity and give an example
21. Define removable singularity, give an example
22.Find the residue of the function f ( z )
4
at a simple pole.
z ( z 2)
3
z2
23.Find the residue of f ( z )
at its simple pole.
(z-1)(z+2)2
sin z
24.Find the residue of f ( z )
at its pole
z4
/15/
PART-B
01.Using Cauchy’s integral formula, evaluate
C is the
circle z
sin z 2 cos z 2
dz where
(z+1) (z+2)
C
3
z
02. Using Cauchy’s integral formula, evaluate
C
(z+1) 2 (z+3)
dz where
where
C is the circle z+1 1
03. Using Cauchy’s integral formula, evaluate
circle
z+1+i
2
04. Using Cauchy’s integral formula, evaluate
circle z-2-i
z dz
where C is
(z-1)(z-2) 2
C
z-1 1 using Cauchy’s Integral formula
z-2
1
using Cauchy’s integral formula
2
3z 2 +7z+1 dz
where C is |z|=2,find f(4),f’(1) and f “(1).
(z-a)
C
07. If f(a)=
08. Find the Laurent’s series expansion of
(i)
z 1 1 (ii) z
09. Expand
0
z
z+1
dz where C is the
z - 4 z 3 +4z 2
C
4
2
2z 2 +z dz
05. Evaluate
where C is
(z 2 -1)
C
06. Evaluate
z+1
dz where C is the
z
+2z+4
C
2
1
z(z-1)
1 and 0
1
valid in the regions
z 2 -3z+2
2
as Laurent’s series valid in the regions
z 1 1
z2 1
10. Expand f(z) = 2
z -5z+6
in a Laurent’s series for
11. Find the Taylor’s series to represent
z
z2 1
in z
(z+2)(z+3)
2 and 2
2
z
in a Laurent’s series for 1 z 2
(z +1)(z 2 +4)
7z 2
1z 1 3
13. Find the Laurent’s series of
in
(z+1)z(z-2)
12. Expand f(z) =
2
z
3
/16/
9z2 4z 1
14. Obtain the Laurent expansion for
in the region
(z-1)(2z-1)(z+2)
1
z 1 3 .Hence find the
residue of the function at z=1.
ez dz
where C is z
(z 2 + 2 ) 2
C
15.Using Cauchy’s Residue theorem, evaluate
4
3z 2 +z-1
dz around the circle z 2
(z 2 -1)(z-3)
C
dz
17. Using Cauchy’s Residue theorem, evaluate
where C is z-i 2
2
(z +4) 2
C
16. Use residue theorem to evaluate
z 1
at its poles and hence
(z+1) 2 (z-2)
f(z) dz ,C is the circle z-i 2
18. Find the residues of
evaluate
f ( z)
C
2z
19. Evaluate
e dz
where C is x 2 +y 2
4
C (z+1)
z3 dz
20. Evaluate
(2z+i)3
C
4
where C is the unit circle
2
21. Using the method of contour integration evaluate
0
2
22. Evaluate
0
d
13+5 sin
2
23. Evaluate
0
sin 2 d
5- 3 cos
using Contour integration.
using Contour integration
2
26. By Contour integration evaluate
0
1+2cos
5+4cos
d
2
27. Using Contour integration ,Prove that
0
2
28. Evaluate
0
29 Evaluate
d
,0 a 1
1-2a cos +a 2
cos 3 d
5- 4 cos
12
d
(a b 0) using Contour integration.
a+b cos
cos mx
dx, using Contour integration
a 2 +x 2
0
30. Evaluate
0
x sin mx
dx, using Contour integration
x2 a2
31. By Contour integration evaluate
x2
dX
(x 2 +a 2 )(x 2 b2 )
0