department of applied science & humanities (mathematics
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Government Engineering College, Nawada
Department of Applied Science & Humanities (Mathematics)
Tutorial Sheet-I Session : 2019-20(Even Sem.) Semester : II
Course/
Branch
: B. Tech./ME Paper Name : Mathematics-II
(102202)
Module : 2 Topic Covered : FIRST ORDER ORDINARY
DIFFERENTIAL EQUATIONS
Name of Faculty: Dr. Rajnesh Kumar
Note: Following are the problems which are required to be done by the students for an overall
understanding of the topics.
1. Find theorder and degree of the following differential equations. State also whether they are linear
or non-linear (i) 1 + dy
dx
2
5/2
=d2y
dx2 (ii) d2y
dx2 + 5dy
dx+ 6y = 0 (iii) yโฒโฒ + xyyโฒ + 3y = 5x (iv)
(yโฒ)2 + 3xyโฒ + y = 0 (v) yโฒ = siny
2. Eliminate the arbitrary constants and obtain the following differential equations satisfied by it
(i) y = a cosฮธx + b sinฮธx, ฮธ: fixed constant (ii) y = c cos(pt โ a), p: fixed constant (iii)
x2 + y2 = a2 (iv) y = 2cx โ c2(v) x2 + y2 โ 2ay = 0.
3. Reduce to separable form and solve the following differential equations (i) (xyโฒ-y) cos(y/x) +x=0
(ii) xyโฒ = eโxy โ y (iii) dy
dx= exโy + x2eโy (iv) x2(1 โ y)dy + y2(1 + x)dx = 0 (v)
sec2x tany dx + sec2y tanx dy = 0 (vi) yโฒ = cos(x + y) + sin(x + y) (vii) sin3xdy
dx= siny.
4. Solve the following differential equations (i)dy
dx=
x2y
x3+y3 (ii) (x2 โ y2)dx โ xy dy = 0 (iii)
(1 + ex/y) + ex/y(1 โ x/y)dy = 0 (iv)( x2 + 4y2 + xy)dx โ x2dy = 0 (v) (3xy + y2) dx +
(x2 + xy)dy = 0.
5. Solve the following differential equations(i)dy
dx=
xโyโ2
2xโ2yโ3 (ii)
dy
dx=
2xโ6y+7
xโ3y+4 (iii)
dy
dx=
x+2yโ3
2x+yโ3 .
6. Solve the following differential equations
(i)(x2 + 1)dy
dx+ 2xy = 4x2 (ii) x
dy
dx+ y = y2 log x(iii) x
dy
dx= 2y + x4 + 6x2 + 2x (iv) (1 +
y2)dx = (tanโ1y โ x)dy (v) (x + 1)dy
dxโ y = e3x(x + 1)2(vi) (1 + x2)
dy
dx+ y = etan โ1x
7. Solve the following differential equations(i) (y2exy 2+ 4x3)dx + (2xyexy2
โ 3y2)dy = 0
(ii)(2x3 + 3y2x โ 7x)dx + (3x2y + 2y3 โ 8y)dy = 0 (iii)(x2y โ 2xy2)dx โ (x3 โ 3x2y)dy =
0 (iv) (1 โ xy)ydx โ x(1 + xy)dy = 0 (v)(xy3 + y)dx + 2(x2y2 + x + y4)dy = 0 (vi) y(xy +
2x2y2)dx + x(xy โ x2y2)dy = 0 (vii) (cosy + ycosx)dx + (sinx โ x siny)dy = 0.
8. Solve: (i) xdy
dx+ y = x3y6 (ii) yโฒ + 4xy + xy3 = 0 (iii) yโฒ โ y = y2(sinx + cosx).
9. Solve : (i) dy
dx
2โ 5
dy
dx+ 6 = 0 (ii)
dy
dx
2+ 2x
dy
dxโ 3x2 = 0 (iii) x4
dy
dx
2โ x
dy
dxโ y = 0.
10. Solve: (i) y = 2px + y2p3 (ii) y = xyโฒ + (yโฒ)2 (iii) y = xyโฒ โ e2yโฒ (iv) y = xyโฒ โ1
yโฒ .
11. Find the orthogonal trajectories of the hyperbolas x2 โ y2 = c.
12. Find the orthogonal trajectories of the family of circles passing through the points (0, 2) and (0, -
2).
13. Find the orthogonal trajectories of the following family of curves (i)r = c(1 + cosฮธ) (ii) r2 =
c sin(2ฮธ).
14. A body is heated to 100โ and placed in air at 20โ . After one hour its temperature is 60โ. How
much additional time is required for it to cool to 30โ.
15. In a radioactive decay, initially 50 mg of the material is present and after two hours, the material
has lost 10% of its original mass. Find the mass at any time t and the half-life of the material.
16. A particle falls down from rest in the air whose resistance is prepositional to the square of the
velocity. Find the velocity as a function of x.
Text / Reference Books:
1. Peter V. O` Neil, A text book of Engineering Mathematics, Thomson (Cengage Learning), 2nd
Edition, 2010.
2. B.S.Grewal, Advanced Engineering Mathematics, Khanna Publishars, 40th
Edition, 2010.
3. E. Kreyszig, โAdvanced Engineering Mathematicsโ, John Wiley and Sons, New York, 2005.
4. B.V. Ramanna, โHigher Engineering Mathematicsโ, Tata Mcgraw Hill Publishing Company Ltd.,
2008.
5. R.K. Jain and S.R.K. Iyengar, โAdvanced Engineering Mathematicsโ, Narosa Publishing House,
2008.
Government Engineering College, Nawada
Department of Applied Science & Humanities (Mathematics)
Tutorial Sheet-II Session : 2019-20(Even Sem.) Semester : II
Course/
Branch
: B. Tech./ME Paper Name : Mathematics-II
(102202)
Module : 3 Topic Covered : ORDINARY DIFFERENTIAL
EQUATIONS OF HIGHER
ORDERS Name of Faculty: Dr. Rajnesh Kumar
Note: Following are the problems which are required to be done by the students for an overall
understanding of the topics.
1. Examine whether the following functions are linearly independent (i) 1, ๐๐๐ ๐ฅ, ๐ ๐๐๐ฅ (ii)
๐๐๐ฅ, ๐๐ ๐ฅ2, ๐๐ ๐ฅ3 (iii) ๐โ๐ฅ , ๐ ๐๐๐ฅ, ๐๐๐ ๐ฅ (iv) ๐ฅ, ๐ฅ2 , ๐ฅ3 (v) ๐๐ฅ , ๐2๐ฅ , ๐3๐ฅ .
2. Find a general solution of the following differential equations: (i) ๐ฆโฒโฒ โ 4๐ฆ = 0 (ii) ๐ฆโฒโฒ โ ๐ฆโฒ โ
2๐ฆ = 0 (iii) ๐ฆโฒโฒ + ๐ฆโฒ โ 2๐ฆ = 0 (iv) ๐ฆโฒโฒ โ 4๐ฆโฒ โ 12๐ฆ = 0 (v) ๐ฆโฒโฒ + 9๐ฆโฒ = 0 (vi) 9๐ฆโฒโฒ โ 12๐ฆโฒ +
4๐ฆ = 0 (vii) ๐ฆโฒโฒ โ ๐ฆโฒ โ 6๐ฆ = 0 (viii) 4๐ฆโฒโฒ + 4๐ฆโฒ + ๐ฆ = 0 (ix) ๐ฆโฒโฒโฒ โ ๐ฆโฒโฒ โ 5๐ฆโฒ + 6๐ฆ = 0 (x)
8๐ฆโฒโฒโฒ โ 12๐ฆโฒโฒ + 6๐ฆโฒ โ ๐ฆ = 0 (xi) ๐ฆ๐๐ฃ โ ๐2๐ฆ = 0 (xii) ๐ฆ๐๐ฃ + 32๐ฆโฒโฒ + 256๐ฆ = 0.
3. Solve the following differential equations: (i) ๐ฆโฒโฒโฒ + ๐ฆ = ๐๐ฅ + 2๐โ๐ฅ (ii) ๐ฆโฒโฒ โ 4๐ฆโฒ + 3๐ฆ =
๐ ๐๐3๐ฅ ๐๐๐ 2๐ฅ(iii) ๐ฆโฒโฒ โ 4๐ฆโฒ + 4๐ฆ = ๐๐ฅ + ๐ ๐๐2๐ฅ (iv) ๐ฆโฒโฒ + 4๐ฆ = ๐๐๐ ๐ฅ ๐๐๐ 3๐ฅ (v) ๐ฆโฒโฒ โ 4๐ฆ = ๐ฅ2
(vi) ๐ฆโฒโฒ โ 2๐ฆโฒ + 3๐ฆ = ๐๐๐ ๐ฅ + ๐ฅ2 (vii) ๐ฆโฒโฒ โ 4๐ฆโฒ + 4๐ฆ = ๐ฅ2 + ๐๐ฅ + ๐๐๐ 2๐ฅ (viii) ๐ฆโฒโฒ โ 2๐ฆโฒ + ๐ฆ =
๐ฅ๐๐ฅ๐ ๐๐๐ฅ (ix) ๐ฆโฒโฒ โ 3๐ฆโฒ + 2๐ฆ = ๐ ๐๐2๐ฅ + ๐ฅ๐๐ฅ .
4. Solve by method of variation of parameters: (i) ๐ฆโฒโฒ + ๐2๐ฆ = ๐ ๐๐ ๐๐ฅ (ii) ๐ฅ2๐ฆโฒโฒ + ๐ฅ๐ฆโฒ โ ๐ฆ = ๐ฅ2๐๐ฅ
(iii) ๐ฅ2๐ฆโฒโฒ โ 4๐ฅ๐ฆโฒ + 6๐ฆ = ๐ ๐๐(๐๐๐ ๐ฅ) (iv) ๐ฆโฒโฒ + 4๐ฆ = ๐๐๐ ๐ฅ (v) ๐ฆโฒโฒ + ๐2๐ฆ = ๐๐๐ ๐๐ax (vi)
๐ฆโฒโฒ + ๐ฆ = ๐ก๐๐ ๐ฅ (vii) ๐ฆโฒโฒ + 6๐ฆโฒ + 9๐ฆ =๐โ3๐ฅ
๐ฅ (viii) ๐ฆโฒโฒ + 4๐ฆโฒ + 4๐ฆ = ๐โ2๐ฅ๐ ๐๐ ๐ฅ (ix) ๐ฆโฒโฒ โ ๐ฆ =
2
1+๐๐ฅ
(x) ๐ฅ2๐ฆโฒโฒ + ๐ฅ๐ฆโฒ โ ๐ฆ = ๐ฅ2๐ฆ (xi) ๐ฅ2๐ฆโฒโฒ + 3๐ฅ๐ฆโฒ + ๐ฆ =1
(1โ๐ฅ)2.
5. Solve the following differential equations:(i) ๐ฅ๐ฆโฒโฒ โ (2๐ฅ โ 1)๐ฆโฒ + (๐ฅ โ 1)๐ฆ = 0 (ii) (1 โ
๐ฅ2)๐ฆโฒโฒ + ๐ฅ๐ฆโฒ โ ๐ฆ = ๐ฅ (1 โ ๐ฅ2 )3/2(iii) ๐ฆโฒโฒ โ ๐๐๐ก ๐ฅ ๐ฆโฒ โ (1 โ ๐๐๐ก ๐ฅ) ๐ฆ = ๐๐ฅ๐ ๐๐ ๐ฅ .
6. Solve: (i) ๐ฆโฒโฒ โ 2๐ก๐๐ ๐ฅ ๐ฆโฒ + ๐ฆ = 0 (ii) ๐ฆโฒโฒ โ 4๐ฅ๐ฆโฒ + (4๐ฅ2 โ 1)๐ฆ = โ3๐๐ฅ2๐ ๐๐ 2๐ฅ (iii) ๐ฅ2๐ฆโฒโฒ โ
2(๐ฅ + ๐ฅ2)๐ฆโฒ + (๐ฅ2 + 2๐ฅ + 2)๐ฆ = 0 (iv) (๐๐๐ ๐ฅ) ๐ฆโฒโฒ + (๐ ๐๐ ๐ฅ) ๐ฆโฒ โ 2(๐๐๐ 3๐ฅ)๐ฆ = 2๐๐๐ 5๐ฅ .
7. Solve: (i) ๐ฅ3๐ฆโฒโฒโฒ + 2๐ฅ2๐ฆโฒโฒ + 2๐ฆ = ๐ฅ +1
๐ฅ (ii) ๐ฅ2๐ฆโฒโฒ โ 5๐ฅ๐ฆโฒ + 3๐ฆ = ๐๐ ๐ฅ (iii) (๐ฅ + 1)2๐ฆโฒโฒ + (๐ฅ +
1)๐ฆโฒ + ๐ฆ = 4๐๐๐ (๐๐๐(1 + ๐ฅ)) (iv) (2๐ฅ + 5)2๐ฆโฒโฒ + 6(2๐ฅ + 5)๐ฆโฒ + 8๐ฆ = ๐ฅ .
8. Find the regular and singular points of the differential equations (i) (1 โ ๐ฅ2)๐ฆโฒโฒ โ 2๐ฅ๐ฆโฒ + ๐(๐ +
1)๐ฆ = 0 (ii) ๐ฅ2๐ฆโฒโฒ + ๐๐ฅ๐ฆโฒ + ๐๐ฆ = 0.
9. Classify the singular points of the following equations (i) ๐ฅ2๐ฆโฒโฒ + (๐ ๐๐ ๐ฅ)๐ฆโฒ + (๐๐๐ ๐ฅ)๐ฆ = 0 (ii)
๐ฅ3(๐ฅ2 โ 1)๐ฆโฒโฒ โ ๐ฅ(๐ฅ + 1)๐ฆโฒ โ (๐ฅ โ 1)๐ฆ = 0 (iii) ๐ฅ2๐ฆโฒโฒ + 2๐ฅ๐ฆโฒ + (๐ฅ2 โ ๐2)๐ฆ = 0.
10. Find the power series solution about ๐ฅ = 0, of the differential equation (i) ๐ฆโฒโฒ โ 2๐ฆ = 0 (ii)
(1 โ ๐ฅ2)๐ฆโฒโฒ โ 2๐ฅ๐ฆโฒ + 2๐ฆ = 0.
11. Find the power series solution about ๐ฅ = 2 of the equation ๐ฆโฒโฒ + (๐ฅ โ 1)๐ฆโฒ + ๐ฆ = 0.
12. Find the series solutions of the following differential equations by the Frobenious method: (i) )
๐ฅ2๐ฆโฒโฒ + 2๐ฅ๐ฆโฒ + (๐ฅ2 โ ๐2)๐ฆ = 0 (ii) 9๐ฅ(1 + ๐ฅ)๐ฆโฒโฒ โ 6๐ฆโฒ + 2๐ฆ = 0 (iii) (1 โ ๐ฅ2)๐ฆโฒโฒ โ 2๐ฅ๐ฆโฒ +
6๐ฆ = 0.
13. Find the series solutions about the indicated point of the following differential equations by the
Frobenious method: (i) 2(1 โ ๐ฅ)๐ฆโฒโฒ โ ๐ฅ๐ฆโฒ + ๐ฆ = 0, x=1 (ii) ๐ฅ(๐ฅ โ 2)๐ฆโฒโฒ + 4๐ฆโฒ + 3๐ฆ = 0, ๐ฅ = 2.
14. Express ๐(๐ฅ) = 3๐3(๐ฅ) + 2๐2(๐ฅ) + 4๐1(๐ฅ) + 5๐0(๐ฅ) as polynomial in ๐ฅ, where ๐๐ (๐ฅ) is the
Legendre polynomial of order ๐.
15. Express ๐(๐ฅ) = ๐ฅ4 + 2๐ฅ3 โ 6๐ฅ2 + 5๐ฅ โ 3 in the terms of Legendre polynomials.
16. Show that (i) ๐๐(1) = 1 (ii) ๐๐(โ๐ฅ) = (โ1)๐ ) ๐๐(๐ฅ) (iii) ๐โฒ๐(1) = ๐(๐ + 1)/2 (iv)
๐๐(๐ฅ) ๐๐ฅ = 01
โ1 (v) ) ๐๐
2(๐ฅ) ๐๐ฅ =2
2๐+1
1
โ1.
17. Show that ๐ฝ๐(๐ฅ) is a even function for n even and an odd function for ๐ odd where n is an integer.
18. Prove that (i) ๐ฝ1/2(๐ฅ) = 2
๐๐ฅ๐ ๐๐ ๐ฅ (ii) ๐ฝโ1/2(๐ฅ) =
2
๐๐ฅ๐๐๐ ๐ฅ (iii) ๐ฝ3(๐ฅ) =
8
๐ฅ2 โ 1 ๐ฝ1(๐ฅ) โ
4
๐ฅ๐ฝ0(๐ฅ) (iv) ๐ฝ5/2(๐ฅ) =
2
๐๐ฅ
1
๐ฅ2 (3 โ ๐ฅ2)๐ ๐๐ ๐ฅ โ3
๐ฅ๐๐๐ ๐ฅ (v) ๐ฝโ5/2(๐ฅ) =
2
๐๐ฅ
1
๐ฅ2 (3 โ ๐ฅ2)๐๐๐ ๐ฅ +
3
๐ฅ๐ ๐๐ ๐ฅ .
Text / Reference Books:
1. Peter V. O` Neil, A text book of Engineering Mathematics, Thomson (Cengage Learning), 2nd
Edition, 2010.
2. B.S.Grewal, Advanced Engineering Mathematics, Khanna Publishars, 40th
Edition, 2010.
3. E. Kreyszig, โAdvanced Engineering Mathematicsโ, John Wiley and Sons, New York, 2005.
4. B.V. Ramanna, โHigher Engineering Mathematicsโ, Tata Mcgraw Hill Publishing Company Ltd.,
2008.
5. R.K. Jain and S.R.K. Iyengar, โAdvanced Engineering Mathematicsโ, Narosa Publishing House,
2008.
Government Engineering College, Nawada
Department of Applied Science & Humanities (Mathematics)
Tutorial Sheet-III Session : 2019-20(Even Sem.) Semester : II
Course/
Branch
: B. Tech./ ME Paper Name : Mathematics-II
(102202)
Module : 4 Topic Covered : Complex Variables-
Differentiation
Name of Faculty: Dr. Rajnesh Kumar
Note: Following are the problems which are required to be done by the students for an overall
understanding of the topics.
1. Check whether the following limits exist or not
(i)z
zz
z
)Im()Re(lim
0โ, (ii)
0limz
z
zโ, (iii)
2
20
Imlimz
z
zโ.
2. Find out whether the following functions are continuous at origin or not, where (0) 0f = ,
(i) 2
2
Re z
z, (ii)
2Im z
z, (iii)
2 1Rez
z
.
3. Examine the differentiability of the following complex valued functions
(i) ( )f z z= at origin, (ii) 2
( )f z z= at 0z = , (iii) 2( )f z z= .
4. Determine the analyticity of the following functions.
(i) zzf sin)( = , (ii) )(2)( 22 yxixyzf ++= , (iii) 2( ) (cosh sinh )xf z e y i y= + .
5. Show that the function ,)(4โโ= zezf z 0 and 0)0( =f is not analytic at the origin even through
CR equation are satisfied.
6. Examine which of the following functions are harmonic:
(i) 23 34),( xyxyxyxu โโ= , (ii) )sincos(),( yyyxeyxu x += โ .
7. Using the C-R equations find the harmonic conjugate of the following functions
(i) yxxyyxu 32222 +โโโ= , (ii) )log(2
1),( 22 yxyxu +=
where, ( )f z u iv= + is an analytic function.
8. Find the analytic function f(z) using Milne โThomson method:
(i) yxu coshcos= (ii) ).4)((),( 22 yxyxyxyxu ++โ=
9. Show that the function ivuzf +=)( , where
=
+
+
=
0,0
0,)(
)( 104
52
z
zyx
iyxyx
zf
Satisfies the Cauchy-Riemann equations at 0=z . Is the function analytic at 0=z ?
10. Deterrmine the analytic function ( )f z u iv= + , if )cosh(cos2
sincos
yx
exxvu
y
โ
โ+=โ
โ
and .0)2( =
f
11. Prove that an analytic function with constant modulus is constant.
12. Find the bilinear transformation which maps the points 2zi,z2,z 321 โ=== into the points
1wi,w1,w 321 โ=== .
13. Discuss the application of the transformation iz
1izw
+
+= to the areas in the z-plane which are
respectively inside and outside the unit circle with its centre at the origin. 14. Find the general homographic transformation which leaves the unit circle invariant. 15. Every bilinear transformation maps circles or straight lines into circles or straight lines.
16. Show that the transformation ,tanh 1 zw โ= maps the upper half of the z -plane conformally on the
strip .2
10 v
17. Show that the resultant (or product) of two bilinear transformations is a bilinear transformation. Explain the concept of isogonal mapping and conformal mapping with illustrative examples.
18. Find the fixed points and the normal of the following bilinear transformation: (i)2โ
=z
zw (ii)
.1
1
+
โ=
z
zw
19. Show that the line x3y = is map onto the circle under the bilinear transformation iz
izw
+
+=
4
2. Find
the centre and radius of image circle. 20. Find the bilinear transformation that maps the point 0zi,z,z 321 === into the points
=== 321 wi,w0,w .
21. Show that the inverse of the point a with respect to the circle R=c-z is the point )(
Rc
2
ca โ+ .
Text / Reference Books: 1. R. K. Jain & S. R. K. Iyengar. โAdvanced Engineering Mathematics,โ Narosa Publishing House Pvt.
Ltd., 3 Ed., 2011
2. J. K. Goyal & K. P. Gupta. โFunctions of a Complex Variable,โ Pragati Prakashan., 17th edition 2003
3. Shanti Narayan & P.K. Mittal. โ Theory of Functions of a Complex Variable,โ S. Chand Revised Ed,
2010
4. Ruel V. Churchill, Complex variables and Applications
5. S. Ponnusamy. โFoundation of Complex Analysis,โ Narosa Publishing House Pvt. Ltd., 2nd Ed. 2005
6. M. R. Spiegel. โTheory of Complex Variable,โ Mc - Graw HiIl Publication, 1981.
7. Murray Spiegel, Seymour Lipschutz, John Schiller & Dennis Spellman. โ Schaum's Outline of
Complex Variables, 2edโ
Government Engineering College, Nawada
Department of Applied Science & Humanities (Mathematics)
Tutorial Sheet-IV Session : 2019-20(Even Sem.) Semester : II
Course/
Branch
: B. Tech./ ME Paper Name : Mathematics-II
(102202)
Module : 5 Topic Covered : Complex Variables-
Integration
Name of Faculty: Dr. Rajnesh Kumar
Note: Following are the problems which are required to be done by the students for an overall
understanding of the topics.
1. If a function )(zf is analytic for finite values of z , and is bounded, then )(zf is constant.
2. Find upper bound of integral โ
+
C
iz
dzz
zLogez
2
)3(2
3
, 3/0,2: == iezzC .
3. What curve is represented by the function 36)2(9)1(4 22 =++โ yx .
4. Find the value of the integral dzixyx
i
+
+โ
21
0
32 )3( along the real axis from 0=z to 1=z and then, along
a line parallel to the imaginary axis from 1=z to iz 21+= .
5. Use ML-inequality to show that 2
1
12
+C
dzz
zwhere C is the straight line segment from 2=z to
iz += 2 .
6. Find the value of the integral izdzzz
e
C
z
4,)2)(1(
2
=โโ .
7. Find the value of the integral dyxy
ydx
yx
x
C
2222 โ+
โ where C is the boundary of the triangle with the
vertices (2, 0), (4, 0) and (4, 3).
8. State and prove fundamental theorem of integral calculus.
9. Evaluate the following integrals by Cauchyโs integral formula
a. dzz
zzc โ
++
2
652
where c is the circle 13 == zandz ,
b. โ
cdz
z
z
2
2
)6
(
sin
where c is the circle 1=z ,
c. โc
dzz 1
1
2 where c is the circle 2=z .
10. If the curve 20),exp()( = titt then find the integral +
+
dzz
z
)1(
12
.
11. If Cwzwfzfwzf +=+ ,)()()( and )(zf is analytic in C , show that czzf =)( , where c is
constant.
12. Prove that non constant entire function is unbounded.
13. Suppose )(zf is entire function such that 2
)( zzf , ifif == )2(,2)1( find )(zf .
14. Find the radius of convergence of the following power series (i)
++
n
1iz
i 2(ii)
n
nz
2
n i2
+.
15. Find the radius of convergence of the following power series:
a.
= +=
0 14)(
nn
nzzf
b. .!
)(
0
=
=
nn
n
n
znzf
16. Find the region of convergence of the series (i)
=1n
n
n log
z (ii) ( )
( )
=
โโ
โโ
1
121
!121
n
nn
n
z.
17. Find the first four terms of the Taylorโs series expansion of the function
)(tan)( 1 zzf โ= about 0=z .
18. Obtain the Taylor and Laurentโs series which represents the function 3)2)(z(z
1z2
++
โ in the regions (i)
2z (ii) 32 z (iii) 3z .
19. Expand )2)(1(
23)(
++
โ=
zzz
zzf in Laurentโs series valid for region .3|2|1 + z
20. Expand )9(
1897)(
3
2
zz
zzzf
โ
โ+= in the region (i) 3||0 z (ii) 3z (iii) 21 z .
21. Find the kind of singularity of the function
โ=
z-1
1sin
2zf(z)
2z.
22. What kind of singularity has the function
(i)
=
z
1cos
1f(z)
at 0=z (ii) ( )1/z cosec
at =z .
23. Discuss the singularity of the following functions:
a. z
ezf
z/1
)( =
b. .11
1cos)( =
โ= zat
zzf
24. Find the poles and residue of the poles of the following function:
)3)(4)(2(
)(4
++โ=
zzz
zzf
25. Determine the nature of the pole at the origin of the function mzz sin
ef(z)
z
= .
26. Show that the function 21/zeโ has no singularities.
27. Find the residue of zcot(z) = at the points n=nz for 1,2,...n = what is the nature of singularity at
=z ? Justify your answer.
28. If an analytic function f(z) has a pole of order m at a=z , then 1/f(z)has a zero of order m at a=z
and conversely.
29. If a=z is an isolated singularity of f(z) and if f(z) is bounded on some deleted neighborhood of a ,
then prove that a is a removable singularity.
30. Find residues of (i)2zz
12z2 โโ
+ (ii)
12
3
โz
z at .=z
31. Evaluate (i) +
2ฯ
0
2
cosba
dsin
(ii) dx
+0
22
2
)1(x
x.
32. By contour integration, prove that 4x
xsin
0
2 =
dx
.
33. Evaluate the following integrals using Cauchy Residue Theorem:
a.
2:;)12()1(
75
2=
+โ
โ zCdz
zz
z
c
b. .2:;)4)(1(
12
2
2
=+โ
+
C
zCdzzz
z
34. Evaluate the following integrals by contour integration:
a. +
2
0 cos45
d
b. .cos45
2cos2
0
โ
d
35. Apply calculus of residue to prove that:
โ=
+โ
2
0
2
2
2,
1
2
cos21
2cos
a
a
aa
d (a2 < 1 ).
36. Evaluate the following integrals by contour integration:
a.
โ + )1( 2x
dx
b.
+022
.)(
sindx
ax
xx
37. Evaluate the following integrals using Cauchy Residue Theorem:
=โโ
+โ
C
zCdzzz
zz1:;
)13()12(
523302
2
.
38. Apply calculus of residue to prove that:
.0);1(2)(
sin
222โ=
+
โ ae
adx
axx
mx ma
C
39. Verify the integral ,sin1x
x
0
-a
adx =
+
10 a .
40. Let f(z) be analytic in Rz
and Mf(z)
onRz =
. Then prove that .R
M(0)'f
Text / Reference Books: 1. R. K. Jain & S. R. K. Iyengar. โAdvanced Engineering Mathematics,โ Narosa Publishing House Pvt.
Ltd., 3 Ed., 2011
2. J. K. Goyal & K. P. Gupta. โFunctions of a Complex Variable,โ Pragati Prakashan., 17th edition 2003
3. Shanti Narayan & P.K. Mittal. โ Theory of Functions of a Complex Variable,โ S. Chand Revised Ed,
2010
4. Ruel V. Churchill, Complex variables and Applications
5. S. Ponnusamy. โFoundation of Complex Analysis,โ Narosa Publishing House Pvt. Ltd., 2nd Ed. 2005
6. M. R. Spiegel. โTheory of Complex Variable,โ Mc - Graw HiIl Publication, 1981.
7. Murray Spiegel, Seymour Lipschutz, John Schiller & Dennis Spellman. โ Schaum's Outline of
Complex Variables, 2edโ