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TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 1
COURSE FILE
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 2
TIRUMALA ENGINEERING COLLEGE
BOGARAM-R.R.DIST. DEPARTMENT OF HUMANITIES&SCIENCES
COURSE FILE
BY
PROF. : P.SHANTAN KUMAR
M.Sc.(Maths).,M.Phil.,B.Ed.,D.Ph.,
SUBJECT : MATHEMATICS-III
BRANCH : common to ECE&EEE branches
YEAR : II-B.TECH – I-SEM-A.Y. 2011-2012
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 3
CONTENTS
ACADEMIC CALENDER
SYLLABUS
TEACHING SCHEDULE
LESSON PLAN
LECTURE NOTES
ASSIGNMENTS(UNIT WISE)
IMPORTANT QUESTIONS (UNIT WISE)
JNTU PREVIOUS YEARS QUESTION PAPERS
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 4
ACADEMIC CALENDER 2011--2012
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
HYDERABAD
II -B.Tech-I-SEM-Common to ECE&EEE Branches II / IV B.Tech./B.Pharm.-I Sem. (Reg.) (2011-12)
** Mid term examinations are to be conducted during both forenoon and afternoon sessions and
they are to be completed within 3 working days as per the schedule given above.
All the midterm examinations shall be of both objective and subjective type as per the academic regulations. Class work shall be
conducted during supplementary examinations period. The attendance of the candidates appearing for supplementary exams shall be counted.
Extra classes may be conducted, if required, subject to a maximum of 64 periods for each subject in a semester.
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 5
SYLLABUS
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 6
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
HYDERABAD
I-Year B.Tech Common to all Branches T: 3 P: 0 C: 3
MATHEMATICS-III
UNIT-I SPECIAL FUNCTIONS-I
Review of Taylor’s series for a real many valued functions, Series Solutions to differential
equations., Gamma and Beta functions-their Properties-Evaluations of Improper Integrals. Bessel
functions-Properties-Recurrence Relations-Orthogonality.
UNIT-II SPECIAL FUNCTIONS-II
Legendre Polynomials-Properties-Rodrigue’s formula-Recurrence Relations-Orthogonality.
Chebycher’s polynomials-Properties-Recurrence Relations-Orthogonality.
UNIT-III FUNCTIONS OF A COMPLEX VARIABLE
Continuity-Differentiability-Analyticity-properties-Cauchy’s-Riemann Conditions, Maxima-
Minima principle, Harmonic and Conjugate harmonic functions-Milne-Thomson method.
Elementary functions, General power z6, Principle value, Logarithmic function.
UNIT-IV COMPLEX INTEGRATION
Line integral-Evaluation along a path and by indefinite integration-Cauchy’s integral theorem-
Cauchy’s integral formula-Generalized integral formula.
UNIT-V COMPLEX POWER SERIES
Radius of convergence-Expansion in Taylor’s series-Maclaurin’s series and Laurent series.
singular point-Isolated singular point-Pole of order m-Essential singularity.(distinguish between
the real analyticity and complex analyticity)
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 7
UNIT-VI CONTOUR INTEGRATION
Residue-Evaluation of Residue by formula and by Laurent series-Residue theorem.
Evaluation of integrals of the Type:
a. Improper real integrals ∫ f(x) dx , ( ∞,-∞) b. ∫ f(cosӨ ,sin Ө)dӨ , (c,c+2π ) c.
∫ eimx
f(x)dx , ( ∞,-∞) d. integrals by indentation.
UNIT-VII CONFORMAL MAPPING
Transformation by e
z ,imz,z
2,z
n(n positive integer),sinz,cosz,z+a/z. Translation, Rotation,
Inversion and Bilinear transformation-Fixed point-Cross ratio-Properties-Invariance of Circles
and Cross ratio-Determination of bilinear transformation mapping 3 given points.
UNIT-VIII ELEMENTARY GRAPH THEORY
Graphs, representation by matrices: Adjacent Matrix-Incident Matrix – Simple, Multiple,
Regular, Complete,Biparitite & Planner graphs-Hamiltonian and Eulerian Circuits-Trees
Spanning Tree-Minimum Spanning Ttree.
TEXT BOOKS:
1. P.B.BHASKARA RAO,RAMA CHARY,BHUJANGA RAO… B.S.P.PUBLICATION
2. C.SHANKARAIAH……..V.G.S.BOOK LINKS
REFERENCES BOOKS:
1. S.Chand
2. Grewal
3. Complex variable by R.V.Churchill.
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 8
TEACHING SCHEDULE
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 9
TIRUMALA ENGINEERING COLLEGE
Dept.of Humanities & Sciences
Teaching Schedule
Name of the faculty: P.Shantan kumar A.Y. : 2011-12
Subject to be handled: Mathematics-III Total No.of hours required:69
Class: II-B.Tech-I-SEM Total No.of hours available:
Unit
Topic to be covered
No. of hours
required
Teaching aids
required if
any
Reference
books/materials
I
Review of Taylor’s series
1
----
B.S.P.PUBLICATION
Series Solutions to
differential equations
1
----
B.S.P.PUBLICATION
Beta , Gamma functions
3
----
B.S.P.PUBLICATION
Properties
2
----
B.S.P.PUBLICATION
Evaluations of Improper
Integrals
2
----
B.S.P.PUBLICATION
Bessel functions,
Recurrence Relations
1
----
B.S.P.PUBLICATION
Properties, Orthogonality.
1
----
B.S.P.PUBLICATION
II
Legendre Polynomials,
Recurrence Relations
3
----
SHANKARAIAH
Properties-Rodrigue’s
formula, Orthogonality.
3
----
SHANKARAIAH
Chebycher’s polynomials,
Recurrence Relations
3
----
SHANKARAIAH
Properties, Orthogonality.
2
---- SHANKARAIAH
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 10
III
Continuity-
Differentiability-
Analyticity-properties
2
----
B.S.P.PUBLICATION
Cauchy’s-Riemann
Conditions, Maxima-
Minima principle,
2
----
B.S.P.PUBLICATION
Harmonic and Conjugate
harmonic functions-
Milne-Thomson method.
2
----
B.S.P.PUBLICATION
Elementary functions,
General power z6,
2
----
B.S.P.PUBLICATION
Principle value,
Logarithmic function.
2
----
B.S.P.PUBLICATION
IV
Line integral-Evaluation
along a path and by
indefinite integration
2
----
S.chand
Cauchy’s integral theorem
2
----
S.chand
Cauchy’s integral
formula-Generalized
integral formula.
3
----
S.chand
V
Radius of convergence-
Expansion in Taylor’s
series-.
2
----
B.S.P.PUBLICATION
Maclaurin’s series and
Laurent series
2
----
B.S.P.PUBLICATION
singular point-Isolated
singular point
1
----
B.S.P.PUBLICATION
Pole of order m-Essential
singularity
2
---- B.S.P.PUBLICATION
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 11
VI
Residue-Evaluation of
Residue by formula and
by Laurent series
2
----
S.chand
Residue theorem,
Problems
2
----
S.chand
Eval’n of integrals of the
Types: a,b,c,d
2
----
S.chand
VII
Transformations in
bilinear transformations
2
----
Grewal
: Fixed point-Cross ratio-
Properties-Invariance of
Circles and Cross ratio
1
----
Grewal
Determination of bilinear
transformation mapping 3
given points
3
----
Grewal
VIII
Graphs, representation by
matrices
1
----
Grewal
Adjacent Matrix-Incident
Matrix – Simple,
Multiple,
3
----
Grewal
Regular,
Complete,Biparitite &
Planner graphs
3
----
B.S.P.PUBLICATION
Hamiltonian and Eulerian
Circuits
2
----
B.S.P.PUBLICATION
Trees Spanning Tree-
Minimum Spanning Tree.
2
----
S.chand
----
Grand
Total
69
----
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 12
LESSON PLAN
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 13
TIRUMALA ENGINEERING COLLEGE
Dept.of Humanities & Sciences LESSON PLAN
Name of the faculty: P.Shantan kumar A.Y.:2011-12
Subject to be handled: MATHEMATICS-III Total No.of hours required: 69
Class: II-B.Tech-I-SEM Total No.of hours available:
S.No. Date Topic to be covered No. of
Periods Remarks
1
04-07-11
Review of Taylor’s
series
1
2
05-07-11
Series Solutions to
differential equations
1
3
06-07-11
Beta , Gamma
functions
1
4
07-07-11
Beta , Gamma
functions
1
5
11-07-11
Beta , Gamma
functions
1
6
12-07-11
Properties
1
7
13-07-11
Properties
1
8
14-07-11
Evaluations of
Improper Integrals
1
9
18-07-11
Evaluations of
Improper Integrals
1
10
19-07-11
Bessel functions,
Recurrence Relations
1
11
20-07-11
Properties,
Orthogonality.
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 14
12
21-07-11
Legendre Polynomials,
Recurrence Relations
1
13
25-07-11
Legendre Polynomials,
Recurrence Relations
1
14
26-07-11
Legendre Polynomials,
Recurrence Relations
1
15
27-07-11
Properties-Rodrigue’s
formula,
Orthogonality.
1
16
28-07-11
Properties-Rodrigue’s
formula,
Orthogonality.
1
17
01-08-11
Properties-Rodrigue’s
formula,
Orthogonality.
1
18
02-08-11
Chebycher’s
polynomials,
Recurrence Relations
1
19
03-08-11
Chebycher’s
polynomials,
Recurrence Relations
1
20
04-08-11
Chebycher’s
polynomials,
Recurrence Relations.
1
21
08-08-11
Properties,
Orthogonality.
1
22
09-08-11
Properties,
Orthogonality.
1
23
10-08-11
Continuity-
Differentiability-
Analyticity-properties
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 15
24
11-08-11
Continuity-
Differentiability-
Analyticity-properties
1
25
15-08-11
Cauchy’s-Riemann
Conditions, Maxima-
Minima principle,
1
26
16-08-11
Cauchy’s-Riemann
Conditions, Maxima-
Minima principle,
1
27
17-08-11
Harmonic and
Conjugate harmonic
functions-Milne-
Thomson method.
1
28
18-08-11
Harmonic and
Conjugate harmonic
functions-Milne-
Thomson method.
1
29
22-08-11
Elementary functions,
General power z6,
1
30
23-08-11
Elementary functions,
General power z6,
1
31
24-08-11
Principle value,
Logarithmic function.
1
32
25-08-11
Principle value,
Logarithmic function.
1
33
26-08-11
Line integral-
Evaluation along a
path and by indefinite
integration
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 16
34
29-08-11
Line integral-
Evaluation along a
path and by indefinite
integration
1
35
30-08-11
Cauchy’s integral
theorem
1
36
31-08-11
Cauchy’s integral
theorem
1
37
01-09-11
Cauchy’s integral
formula-Generalized
integral formula.
1
38
02-09-11
Cauchy’s integral
formula-Generalized
integral formula.
1
39
03-09-11
Cauchy’s integral
formula-Generalized
integral formula.
1
40
12-09-11
Radius of
convergence-
Expansion in Taylor’s
series-.
1
41
13-09-11
Radius of
convergence-
Expansion in Taylor’s
series-.
1
42
14-09-11
Maclaurin’s series and
Laurent series
1
43
15-09-11
Maclaurin’s series and
Laurent series
1
44
19-09-11
singular point-Isolated
singular point
1
45
20-09-11
Pole of order m-
Essential singularity
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 17
46
21-09-11
Pole of order m-
Essential singularity
1
47
22-09-11
Residue-Evaluation of
Residue by formula
and by Laurent series
1
48
26-09-11
Residue-Evaluation of
Residue by formula
and by Laurent series
1
49
27-09-11
Residue theorem,
Problems
1
50
28-09-11
Residue theorem,
Problems
1
51
29-09-11
Eval’n of integrals of
the Types: a,b,c,d
1
52
03-10-11
Eval’n of integrals of
the Types: a,b,c,d
1
53
04-10-11
Transformations in
bilinear
transformations
1
54
05-10-11
Transformations in
bilinear
transformations
1
55
06-10-11
: Fixed point-Cross
ratio-Properties-
Invariance of Circles
and Cross ratio
1
56
11-10-11
Determination of
bilinear transformation
mapping 3 given
points
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 18
57
11-10-11
Determination of
bilinear transformation
mapping 3 given
points
1
58
12-10-11
Determination of
bilinear transformation
mapping 3 given
points
1
59
13-10-11
Graphs, representation
by matrices
1
60
17-10-11
Adjacent Matrix-
Incident Matrix –
Simple, Multiple,
1
61
18-10-11
Adjacent Matrix-
Incident Matrix –
Simple, Multiple,
1
62
19-10-11
Adjacent Matrix-
Incident Matrix –
Simple, Multiple,
1
63
20-10-11
Regular,
Complete,Biparitite &
Planner graphs
1
64
24-10-11
Regular,
Complete,Biparitite &
Planner graphs
1
65
25-10-11
Regular,
Complete,Biparitite &
Planner graphs
1
66
26-10-11
Hamiltonian and
Eulerian Circuits
1
67
27-10-11
Hamiltonian and
Eulerian Circuits
1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 19
68
28-10-11
Trees Spanning Tree-
Minimum Spanning
Ttree.
1
69
29-10-11
Trees Spanning Tree-
Minimum Spanning
Ttree.
1
70
Signature of the faculty Signature of the H.O.D.
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 20
ASSIGNMENTS(UNIT WISE)
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 21
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-I(COMMON TO ECE & EEE)
1. Solve in Series (1-x2)y11 – xy1 + 4y = 0
2. Solve ∫ (8 – x3)-1/3 dx in between 0 to 2
3. Solve ∫ xm (logx)n dx in between 0 to 1
4. P.T. Г(m) Г(m+1/2) = √π/22n-1 Г(2m)
5. Find the Values of ∫ sin8xdx , ∫ cos6xdx in between 0 to π/2
6. P.T. ∫ (x-a)m (b – x)n dx = (b-a)m+n β(m+1,n+1) in between a to b
7. P.T. ∫ x Jn(αx) Jn(βx) dx = 0 , α ≠β
1/2 [Jn+1(x)]2, α=β in between 0 to 1
8. S.T. d/dx [x-n Jn(x)] = -x-n Jn+1(x)
9. Find the Value of J5 in terms of J0 & J1
10. P.T. J02 + 2[J1
2+J22+……] = 1
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 22
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-II(COMMON TO ECE & EEE)
1. State & Prove Rodrigue’s Formulae
2. P.T. (2n+1)Pn(x) = P1n+1(x) – P1
n-1(x)
3. Express f(x) = x3 -5x2+x+2 in terms of Legendre’s Polynomials
4. P.T. P1n+1+ P1
n = P0 + 3P1+5P3+…….+(2n+1)Pn
5. P.T. ∫ (1 – x2) P1m P1
n dx = 0, if m≠n in between -1 to 1
6. S.T. (2n+1)xPn(x) = (n+1) Pn+1(x) + n Pn-1(x)
7. State and Prove Generating Function for Tn(x)
8. Express the Polynomial 16x4+12x3+6x2+4x -1 in terms of Tn(x)
9. P.T. Tn+1(x) – 2xTn(x) + Tn-1(x) = 0
10. P.T. (i) Tn(1) = 1 (ii) Tn(-1) = (-1)n
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 23
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-III(COMMON TO ECE & EEE)
1. S.T. the function given by f(z) = x3(1+i) – y3(1-i)/(x2+y2), z≠0
= 0 , z=0
Is continuous at the origin
2. Find whether f (z) = x-iy/x2+y2 is analytic or not.
3. P.T. the family of curves u(x,y) = c1 cut orthogonally the family of curves u(x,y) = c2
4. If f (z) = u+iv is an analytic function of z and u-v=ex(cosy-siny). Find f(z) in terms of z
5. Determine the analytic function f(z) = u+iv, given that 3u+2v = y2-x2+16x
6. Find the analytic function whose real part is y+excosy
7. If f(z) is analytic function of z, P.T. І ∂2/∂x2 + ∂2/∂y2І Іf(z)І2 = 4 Іf1(z)І2
8. Separate the Real and Imaginary part of Tanhz.
9. Find the principle values of (1+i)(1-i)
10. Determine the value of sinz = i
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 24
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-IV(COMMON TO ECE & EEE)
1.Evaluate ∫( 3x2+4xy+ix
2)dz along the curve y = x
2 in between (0,0) to (1,1)
2.Evaluate ∫( x-y+x2)dz along the closed path bounded by the curves y = x
2 & y=x in
between z=0 to 1+i
3. Evaluate z+1 dz , C :│z+1+i│= 2 , Cauchy’s Integral Formulae.
z2+2z+4
4. Evaluate z3e
-z dz , C :│z-1│= 1/2 , Cauchy’s Integral Formulae.
(z-1)3
5. Evaluate z -1 dz , C :│z-i│= 2 , Cauchy’s Integral Formulae.
(z+1)2(z-2)
6.Evaluate SinΠz2 +CosΠz
2 dz , C :│z│= 3 , Cauchy’s Integral Formulae.
(z-1)(z-2)
7.Evaluate 4-3z dz , C :│z│= 3/2 , Cauchy’s Integral Formulae.
z(z-1)(z-2)
8.Evaluate e2z
dz , C :│z-1│= 1 , Cauchy’s Integral Formulae.
(z+1)4
9.Evaluate z3-sin3z dz , C :│z│= 2 , Cauchy’s Integral Formulae.
(z-π/2)3
10.Evaluate dz dz , C :│z│= 2 , Cauchy’s Integral Formulae.
Z8(z+4)
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 25
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
ASSIGNMENT QUESTIONS (COMMON TO ECE & EEE)
UNIT-I
1. Solve in Series (1-x2)y11 – xy1 + 4y = 0
2. P.T. Г(m) Г(m+1/2) = √π/22n-1 Г(2m)
3. S.T. d/dx [x-n Jn(x)] = -x-n Jn+1(x)
4. P.T. ∫ x Jn(αx) Jn(βx) dx = 0 , α ≠β
1/2 [Jn+1(x)]2, α=β in between 0 to 1
UNIT-II
1. State & Prove Rodrigue’s Formulae
2. State and Prove Generating Function for Tn(x)
3. P.T. (i) Tn(1) = 1 (ii) Tn(-1) = (-1)n
4. P.T. (2n+1)Pn(x) = P1n+1(x) – P1
n-1(x)
UNIT-III
1. S.T. the function given by f(z) = x3(1+i) – y3(1-i)/(x2+y2), z≠0
= 0 , z=0
Is continuous at the origin
2. If f(z) is analytic function of z, P.T. І ∂2/∂x2 + ∂2/∂y2І Іf(z)І2 = 4 Іf1(z)І2
3. Find the principle values of (1+i)(1-i)
4. Determine the analytic function f(z) = u+iv, given that 3u+2v = y2-x2+16x
UNIT-IV
1. Evaluate z3-sin3z dz , C :│z│= 2 , Cauchy’s Integral Formulae.
(z-π/2)3
2. Evaluate dz dz , C :│z│= 2 , Cauchy’s Integral Formulae.
Z8(z+4)
3. Evaluate ∫( 3x2+4xy+ix
2)dz along the curve y = x
2 in between (0,0) to (1,1)
4. Evaluate ∫( x-y+x2)dz along the closed path bounded by the curves y = x
2 & y=x in
between z=0 to 1+i
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 26
TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
FOR MID QUESTIONS (COMMON TO ECE & EEE)
UNIT-I
1. Solve ∫ (8 – x3)-1/3 dx in between 0 to 2
2. Solve ∫ xm (logx)n dx in between 0 to 1
3. Find the Values of ∫ sin8xdx , ∫ cos6xdx in between 0 to π/2
4. P.T. ∫ (x-a)m (b – x)n dx = (b-a)m+n β(m+1,n+1) in between a to b
5. Find the Value of J5 in terms of J0 & J1
6. P.T. J02 + 2[J1
2+J22+……] = 1
UNIT-II
1.P.T. (2n+1)Pn(x) = P1n+1(x) – P1
n-1(x)
2.Express f(x) = x3 -5x2+x+2 in terms of Legendre’s Polynomials
3.P.T. P1n+1+ P1
n = P0 + 3P1+5P3+…….+(2n+1)Pn
4.P.T. ∫ (1 – x2) P1m P1
n dx = 0, if m≠n in between -1 to 1
5.Express the Polynomial 16x4+12x3+6x2+4x -1 in terms of Tn(x)
6.P.T. Tn+1(x) – 2xTn(x) + Tn-1(x) = 0
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UNIT-III
1. Find whether f (z) = x-iy/x2+y2 is analytic or not.
2. P.T. the family of curves u(x,y) = c1 cut orthogonally the family of curves u(x,y) = c2
3. If f (z) = u+iv is an analytic function of z and u-v=ex(cosy-siny). Find f(z) in terms of z
4. Find the analytic function whose real part is y+excosy
5. Separate the Real and Imaginary part of Tanhz.
6. Determine the value of sinz = i
UNIT-IV
1. Evaluate z+1 dz , C :│z+1+i│= 2 , Cauchy’s Integral Formulae.
z2+2z+4
2. Evaluate z3e
-z dz , C :│z-1│= 1/2 , Cauchy’s Integral Formulae.
(z-1)3
3. Evaluate z -1 dz , C :│z-i│= 2 , Cauchy’s Integral Formulae.
(z+1)2(z-2)
4.Evaluate SinΠz2 +CosΠz
2 dz , C :│z│= 3 , Cauchy’s Integral Formulae.
(z-1)(z-2)
5.Evaluate 4-3z dz , C :│z│= 3/2 , Cauchy’s Integral Formulae.
z(z-1)(z-2)
6.Evaluate e2z
dz , C :│z-1│= 1 , Cauchy’s Integral Formulae.
(z+1)4
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II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-V(COMMON TO ECE & EEE)
1.State &Prove Taylor’s Theorem.
2.State & Prove Laurent’s Theorem.
3.Find the Taylor’s expansion of f(z) = (2z3+1) / (z
2+z) about the point z= i
4.a) Expand f(z) = sinz in a Taylor’s series about z = π/4
b)Expand f(z) = e2z
/ (z-1)3 as Laurent series about the singular point z=1
5.Expand f(z) = 1/ (z2-z-6) about i) z=-1 ii) z=1
6.Expand as a Taylor’s series in f(z) =(2z3+1)/(z
2+1) about z=1
7.Expand f(z) = (z+3) / z(z2-z-2) in power of z
when i) І z І < 1 ii) 1 < Іz І < 2 iii) І z І > 2
8.Find the Laurent series of (z)/(z-1)(z-3) about z=3
9. Find the Laurent series of (ez)/(z)(1-3z) about z=1
10. Find the Laurent series of (z2-1)/(z+2)(z+3)
about i) І z І < 2 ii) 2 < І z І < 3 iii) І z І > 3
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TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-VI(COMMON TO ECE & EEE)
1. Determine the poles & residues of f(z)= (z+1)/z2(z-2)
2. Determine the poles & residues of f(z)= (2z+1)2/(4z
3+z)
3. Determine the residues of f(z) = z2/(z
4+1) of the circle IZI =2
4. Evaluate ∫(4-3z)/ z(z-1)(z-2) dz , where the circle IZI= 3/2 using the residue
theorem.
5. Evaluate ∫ (ez)/ (z
2+ 2 )
2 dz , where the circle IZI= 4 using the residue
theorem.
6. Evaluate ∫ (12z-7)/(12z+3)(z-1)2 dz , where the circle x
2+y
2 =4 using the
residue theorem.
7. Evaluate ∫ (z-3)/ (z2+2z+5) dz ,
where the circle i) IZI= 1 ii) IZ+1-iI=2 iii)IZ+1+iI =2
8. Evaluate ∫ (sin π z2+cos π z
2) / (z-1)
2(z-2) dz , C:IZI =3
9. Prove that ∫dθ /(1+a2-2a cos θ ) = 2a π /(1-a
2) , o<a<1
10. Evaluate ∫ dx / (x4+a
4)
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TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-VII(COMMON TO ECE & EEE)
1. Find the image of IZ-2iI =2 under the transformation w= 1/z
2.Find the image of infinite strip 0<y<1/2 , w=1/z.
3.If w = (1+iz)/(1-iz) , find the image of IZI<1.
4.Show that the transformation w = (2z+3)/(z-4) changes the circle x2+y
2-4x = 0
into the straight line 4u+3=0
5.Find the image of rectangle with vertices (0,0),(2,0),(2,1),(0,1) under the
transformation w = z+(1+2i).
6.Under the transformation w (z-i)/(1-iz) ,
find image of the circle i) IwI=1 ii) IzI=1
7.Show that the transformation w = z +1/z converts the straight line
arg z = a (IaI < π /2 ) into a branch of the hyperbola of eccentricity sec a
8.Find the bilinear transformation which maps the points (-1,0,1) into the points
(0,i,3i).
9. Find the bilinear transformation which maps the points (0,1,∞ ) into the points
(- 1,-2,-i).
10. Find the bilinear transformation which maps the points (-1,i,1) into the points
(-i,0,i).
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TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
QUESTION BANK ON UNIT-VIII(COMMON TO ECE & EEE)
1.Define Directed Graph, Undericted Graph & Complete Graph with examples.
2.Define Biparitite Graph,Planar Graph with examples.
3.Define Adjacency matrix ,incidence matrix.
4.Define Eulers circuit , Hamiltonian Graph.
5.Define Binary tree & spanning tree.
6.Draw the graphs of Hamitonian path & circuits.
7.Draw the graphs of Eulers path & circuits.
8.Write properties of adjacency matrix.
9.Describe the Graph Theory.
10.Describe about in degree & out degree and give with suitable examples.
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TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
ASSIGNMENT QUESTIONS (COMMON TO ECE & EEE)
UNIT-V
1. State & Prove Laurent’s Theorem.
2. Find the Laurent series of (z2-1)/(z+2)(z+3)
about i) І z І < 2 ii) 2 < І z І < 3 iii) І z І > 3
3. Expand f(z) = (z+3) / z(z2-z-2) in power of z
when i) І z І < 1 ii) 1 < Іz І < 2 iii) І z І > 2
4. a) Expand f(z) = sinz in a Taylor’s series about z = π/4
b)Expand f(z) = e2z
/ (z-1)3 as Laurent series about the singular point z=1
UNIT-VI
1. Evaluate ∫ (ez)/ (z
2+ 2 )
2 dz , where the circle IZI= 4 using the residue
Theorem
2. Prove that ∫dθ /(1+a2-2a cos θ ) = 2a π /(1-a
2) , o<a<1
3. Evaluate ∫ (z-3)/ (z2+2z+5) dz ,
where the circle i) IZI= 1 ii) IZ+1-iI=2 iii)IZ+1+iI =2
4. Determine the poles & residues of f(z)= (2z+1)2/(4z
3+z)
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UNIT-VII
1. Show that the transformation w = z +1/z converts the straight line
arg z = a (IaI < π /2 ) into a branch of the hyperbola of eccentricity sec a
2. Find the bilinear transformation which maps the points (-1,i,1) into the points
(-i,0,i).
3. Find the image of rectangle with vertices (0,0),(2,0),(2,1),(0,1) under the
transformation w = z+(1+2i).
4. If w = (1+iz)/(1-iz) , find the image of IZI<1.
UNIT-VIII
1.Define Directed Graph, Undericted Graph & Complete Graph with examples.
2.Define Binary tree & spanning tree.
3.Describe about in degree & out degree and give with suitable examples.
4.Define Adjacency matrix ,incidence matrix.
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TIRUMALA ENGINEERING COLLEGE
II-B.TECH –I-SEM- MATHEMATICS-III
FOR MID QUESTIONS (COMMON TO ECE & EEE)
UNIT-V
1.State &Prove Taylor’s Theorem.
2.Find the Taylor’s expansion of f(z) = (2z3+1) / (z
2+z) about the point z= i
3.Expand f(z) = 1/ (z2-z-6) about i) z=-1 ii) z=1
4.Expand as a Taylor’s series in f(z) =(2z3+1)/(z
2+1) about z=1
5.Find the Laurent series of (z)/(z-1)(z-3) about z=3
6. Find the Laurent series of (ez)/(z)(1-3z) about z=1
UNIT-VI
1. Determine the poles & residues of f(z)= (z+1)/z2(z-2)
2. Deermine the residues of f(z) = z2/(z
4+1) of the circle IZI =2
3. Evaluate ∫(4-3z)/ z(z-1)(z-2) dz , where the circle IZI= 3/2 using the residue
theorem.
4. Evaluate ∫ (12z-7)/(12z+3)(z-1)2 dz , where the circle x
2+y
2 =4 using the
residue theorem.
5. Evaluate ∫ (sin π z2+cos π z
2) / (z-1)
2(z-2) dz , C:IZI =3
6. Evaluate ∫ dx / (x4+a
4)
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UNIT-VII
1. Find the image of IZ-2iI =2 under the transformation w= 1/z
2.Find the image of infinite strip 0<y<1/2 , w=1/z.
3.Show that the transformation w = (2z+3)/(z-4) changes the circle x2+y
2-4x = 0
into the straight line 4u+3=0
4.Under the transformation w (z-i)/(1-iz) ,
find image of the circle i) IwI=1 ii) IzI=1
5.Find the bilinear transformation which maps the points (-1,0,1) into the points
(0,i,3i).
6. Find the bilinear transformation which maps the points (0,1,∞ ) into the points
(- 1,-2,-i).
UNIT-VIII
1.Define Biparitite Graph,Planar Graph with examples.
2.Define Eulers circuit , Hamiltonian Graph.
3.Draw the graphs of Hamitonian path & circuits.
4.Draw the graphs of Eulers path & circuits.
5.Write properties of adjacency matrix.
6.Describe the Graph Theory.
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IMPORTANT QUESTIONS
(UNIT WISE)
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TIRUMALA ENGINEERING COLLEGE,BOGARAM.
Mathematics-III (Code. No:53007 )
Name of the Faculty : P. Shanthan Kumar, M.Sc., M.Phil., B.Ed., D.Ph.,
UNIT-I(Descriptive questions)
1.)Show that (m, n) = ( 1)!( 1)!
( 1)!
m n
m n
for m , n are positive integers.
2.) Show that )2/1(
3.) Prove that J n (x) and J n (x) are linearly dependent.
4.) Show that Relation between and functions is (p, q) = ( ) ( )
( )
p q
p q
5. Prove that d / dx )(xp
p Jx = x p J 1p (x)
6.) Prove that d / dx )(xJ p = J 1p (x) – P/x J p (x)
7.) Prove that. J )(1 xp= p/x J p (x) - J 1p (x)
8.) Express x 3 + 2x 2 -x-3 interms of Lengendre Polynomials.
9.) J 2 -J 0 = 2j11
0
10.) Prove that
1
20 1
nx dx
x = 2.4.6.8......( 1)
1.3.5.7.....
n
n
where n is an odd integer
11.) Show that
x
nx0
1 J n (x) dx = x )(1
1 xJ n
n
12.) dxexdxex xx
0
2
0
42
2 using functions
13.) Show that 2/))4/3()4/1((tan
2/
0
d
14.)Evaluate dxxax
0
224 using - functions
15.) Write dxex axn 2
0
12
interms of functions
16.)
d2/
0
cot using function
17.)
0
2/72
xe xdx using
function
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18.) Evaluate
2
0
)3 3/1
8( dxx using - functions
19.) Prove that
0
/ mye = m (m)
20.) . Prove that J )(2/5 x = x/2 (3-x 2 / x 2 sin x – 3/x cos x)
21.) Prove that d dx 2
1
2
nn JJ = 2 2
1
2/)1(/ nn xJnxJn
22.) J n (x)= (-1) n J n (x)
23.) Express J 3 and J 4 in terms of J 0 and J 1 .
24.) dxxx
1
0
3 1 using functions
25. P.T. i) ∫ e-x2
x7/2
dx = (5/32) Г(1/4) ii) Г[(n+1)/2] = √Π /22n
Г(2n+1)/Г(n+1)
26. P.T. i) Г(n)Г(1-n) = Π/sinnΠ ii) ∫ sin2θ cos
4θ dθ = Π/32
27. P.T. i) ∫ √tanθ dθ = [Г(1/4)Г(3/4)]/2 ii) Find ∫ (8-x3)
1/3 dx
28. P.T. i) J1/2 = √(2/Πx) sinx ii) J2
0 +2(J2
1+j2
2+ ………..) = 1
29. P.T. i) J11
o = (1/2) [J2-J0] ii)(1-2xz+z2)
-1/2 = Σ Pn(x) Z
n
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UNIT--II (D.Q)
1.) Write Legendre duplication formula for gamma function.
2.) Write the general solution of Legendre’s equation.
3.) Show that )12/()1(2))(1( 2
1
1
12
nnndxpx n
4.) Prove that )32)(12/()1(21 1
1
1
2
nnnndxppx nn
5.) Show that 2p 3 (x) 3p 1 (x) = 5x
3
6.) (1-x 2 )p n
1 = (n+1) (x p n - p 1n )
7.) If f(x) = x3+2x
2-x-3 , then find interms of legendre’s polynomials.
8.) (1-x2)P
1n = (n+1) [xPn –Pn+1]
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UNIT—III--A (D.Q)
1. Define continuity of a complex function and Prove that If f(z) is continuous
some neighbor hood of z and differentiable at z = a + Ib then the first order partial
derivatives satisfy the equations yvxu // and xvyu // at point z(
Cauchy Riemann
Equations)
2. Find the polar form of Cauchy Riemann equations
3. Prove that real and imaginary parts of an analytic function are harmonic.
4. Define harmonic conjugate and prove that the family of curves u(x, y) =
constant cuts orthogonally the family of curves v(x, y) = constant if w = f(z) is
analytic
5. If u = x 22 y v = -y/ (x 2 +y 2 ) then show that both u and v are harmonic but u
+iv is not analytic
6. Show that the function defined by f(z) = x 3 (1+i) -y 3 (1-i) / (x 2 +y 2 ) at z 0 and
f(0) = 0 is continuous and satisfies C.R. equations at the origin but f 1 (0) does not
exist.
7. Show that f(z) =
z is not analytic any where.
8. Find the analytic function whose real part is e x ( x cos y + y sin y)
9.. Show that f(x, y) = x 3 -x y 3 +x y +x + y can be the imaginary part of an analytic
function of z = x + i y
10. If f(z) = u + iv is an analytic function of z show that
( x / )(zf ) 2 + 2)(/ zfy = 21 zf
11. Find an analytic function f(z) such that Re )(1 zf = 3x 2 - uy – 3y 2
and f (1+i) = 0
12. Show that function u = 2log (x 2 +y 2 ) is harmonic and find its
harmonic conjugate.
13. Find whether f(z) = x-iy / (x 2 +y 2 ) is analytic or not
14. If w = u +iv is an analytic function of z and u +v = sin2x /
(cosh2y – cos2x) then find f(z).
15. Determine the analytic function f(z) = u +iv given that 3u + 2v =
y 2 -x 2 +16x
16.) Find the analytic function f(z) such that real part of it is
e x2 (xcos2y – ysin2y)
17.) If arg f(z) = constant then prove that f(z) = u + iv is a constant
function.
18. If f is analytic Show that f 1 = rf /sincos
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19. Define harmonic function find the orthogonal trajectories of the family of
curves r 2 cos2 c
20. If f(z) is an analytic function, then Show that log 2zf is a harmonic
function.
21. If u is a harmonic function show that w = u 2 is not harmonic function unless
u is a constant.
22. Show that f(z) = x y 2 (x +iy) / (x 2 + y 4 ) if z 0
= 0 if z = 0 is not analytic at z = 0
although C.R equations are satisfied at origin.
23. Find k such that f(x, y) = x 3 + 3kxy 2 may be harmonic and find its
conjugate.
24. Find the orthogonal trajectories of the family of curves x 3 y – x y 3 = c
25. Find the harmonic conjugate of u = e22 yx cos2xy.
Hence find f(z) in terms of z
26. Show that for the function f(z) = (x y) 2/1 the C.R equations are satisfied at
the origin but f 1 (z) Does not exist.
27. If f = u + iv is analytic in a Domain D and uv is constant in D then Prove
that f(z) is constant
28. Prove that if v is harmonic conjugate of u and u is harmonic conjugate of v
then f(z) is constant.
29. Find the analytic function whose real part is sin/1 rr
30.) Show that the real and imaginary parts of an analytical function
f(z) = u(r, ) + iv(r, ) satisfy Laplace equation in Polar form
2
2
u
r
+
1
r
v
r
+ 2
1
r
2
2
u
= 0
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UNIT--III --B(D.Q)
1) Find all solutions of
ez= 3 + 4i
2) Find the real and imaginary parts of tan-1(x+i y)
3) If (x+iy)1/3= (a+ib) prove that 4(a2-b2) =
4)Separate the real and imaginary parts of log sin z
5) Find the Principal values of
6) Solve for z if Cosh z= -1
7) Find the principal value of
8) Show that tan z= z has only real roots.
9)
10) Find the real and imaginary parts of
11) Prove that = and
=
12) Find the derivative of tanh z
13)Find the modulus and argument of
14) Find the modulus and argument of
15) Determine the real and imaginary parts of log( sin(x+iy))
16)If ,sincos)sin( ii then prove that 4cos = 2sin
17)If Cosec ivui )4
(
prove that u2+v
2=2(u
2-v
2)
18)If tan log(x+iy) =a +i b then prove that )log(tan1
2 22
22yx
ba
a
19) Find all roots of the equation cos z = ½
20)If sincos)tan( ii then prove that 42
n
And )24
tan(log2
1
21)Find the principal and general values of log2
3 i
22) Find the principal value of log ii
23)If cos irei )(
Then prove that )sin(
)sin(2
e
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24)Show that coss and cosh z are even functions while tan z and tanhz are odd functions.
25)Find all the zeros of tan z cot z,tanhz and cothz.
26)Find the real and imaginary parts of tan-1(cos+isin)
27) Solve the equation e2z-1 =1+i
28)Find the general and principal values of loge(-1)
29)Find the principal value of ilog(1+i)
30)solve the equation sin z= cosh 4.
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UNIT--IV(D.Q)
Evaluate the following integrals where z is a complex number:
1)
2)
3)
4)
5)
6)
7)
8) Evaluate around the square with vertices at (0,0), (1,0), (1,1), (0,1).
9)Evaluate where c is the upper half of the circle
10) Evaluate where
And c is the arc from z = -1-i to z = 1+i of the curve y=x3.
11)Evaluate where c is the shortest path from 1+i to 3 +2i .
12) where c is the straight line segment from z=0 to z= 1+2i .
13) where c is is the circle .
14)Evaluate along the line y = x.
15)Evaluate along the parabola x= 2t , y= t2+ 3
joining the points (0,3) and (2,4).
16)Evaluate where c is the curve y=2x2 from (1,2) to (2,8).
17) Evaluate along the line from 0 to 1+i.
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18)Evaluate where c is .
19)Prove that where c is .
20)State and prove Cauchy’s Integral Theorem for a complex function f(z) which
is analytic in a simply connected domain.
21) Verify Cauchy’s Integral theorem for f(z)= z2 taken over the boundary of a
square with vertices at in a counter-clockwise direction.
Using Cauchy’s Integral theorem evaluate where
22)
23)
24)
Evaluate using Cauchy’s Integral Theorem
25) dz where c is
26) where c is
27) where c is a closed curve enclosing z=1.
Evaluate using Cauchy’s Integral Formula
28) where c is
If dz and c is the contour 16x2+9y
2=144
29) find F(2)
30)F’(i)
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TIRUMALA ENGINEERING COLLEGE,BOGARAM.
Mathematics-III (Code. No:53007 )
Name of the Faculty : P. Shanthan Kumar, M.Sc., M.Phil., B.Ed., D.Ph.,
Unit-5 (Descriptive questions)
1.) state and prove Taylor’s theorem
2.) State and prove Laurent’s theorem.
3.) Expand log z about z=1.
4.) Expand f(z) = 1
( 1)( 2)z z about i) z=1 ii) z= -1
5.) For the function f (z) =
32 1
( 1)
z
z z
find Taylor’s series valid in nbd of z = 1.
6.) Expand f (z) = ( 1)
ze
z z about z=2.
7.) Show that when 1z <1 , 2z = 1+
1
( 1)( 1)n
n
n z
8.) Obtain all the Laurent series of the function 7 2
( 1) ( 2)
z
z z z
about z= -1.
9.) Obtain all the Laurent series of the function
2( 1)
( 2)( 3)
z
z z
if
2< z <3.
10.) Find the Laurent series of the function f (z) = 1
( 1)( 2)z z about
z = -2.
11.) Find the Laurent series of the function f (z) = 2
1
4 3z z
about z = -2.
12.) Expand f (z) =
2
2( 1)
ze
z about z=1 as a Laurent series. Also find
the region of convergence.
13.) Express f (z) = ( 1)( 3)
z
z z in a series of positive and negative
powers of (z-1).
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14.) Expand 3
2
(2 1)z about 1) z=0 2) z=2.
15.) Expand sin z in powers of z.
16.) Expand 3
1 cos z
z
about z=0.
17.) Expand f (z) = 2
1
(1 )z z about z=1 as a Laurent series. Also find
the region of convergence.
18.) Expand 2 sinz z about z=a>0.
19.) Find Meclaurin’s series of 2
1
1 z z
20.) Express f (z) = 2 2( 1)( 4)
z
z z as a series for 1< z <2. .
21.) Expand z21/ ze in powers of z.
22.) Find the Meclaurin’s expansion of the function
2( 1)
( 2)( 3)
z
z z
if z >3.
23.) Expand log (1+z) about z = 0 as a Taylor’s series.
24.) Expand sin z in Taylor’s series about z=4
.
25.) Expand the function f (z) = 2
1
3 2z z in the region
0< 1z <1.
26.) Expand log1
1
z
z
in a Meclaurin’s series about z=0.
27.) Expand the function f(z)= 2
1
3 2z z in the region 1< z <2.
28.) Prove that sin 2z =
6 10 142 .....,
3! 5! 7!
z z zz z <
29.) Expand the function f(z)= 2
1
3 2z z in the region z >2.
30.) Expand ( 2)( 2)
( 1)( 4)
z z
z z
if z <1.
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Unit-6 (Descriptive questions)
1. Define Residue and state and prove residue theorem.
2. Find the poles of the function f(z) = 2
( 1)
( 2)
z
z z
and the corresponding residues at each pole.
3. Find the poles of the function
2
2 2
2
( 1) ( 4)
z z
z z
and residues at each pole.
4. Find the poles and residues of the function
2
3
(2 1)
(4 )
z
z z
at each pole.
5. Find the residue of
2
4( 1)
z
z at those poles which lie inside the circle z =2.
6. Evaluate 2
( 3)
( 2 5)
z dz
z z
where the circle 1z i =2.
7. Evaluate
(1 )
( cos sin )
ze dz
z z z
where C is z =1 by residue theorem.
8. Evaluate
(4 3 )
( 1)( 2)
zz dz
z z z
where C is z =3/2 by residue theorem.
9. Evaluate
2
3( 1)
ze dz
z where C is z =2 by residue theorem.
10. Evaluate
sin
cos
zdz
z z where C is z = by residue theorem.
11. Find the poles and residues at each pole for 3( 1)
zze
z .
12. Find the poles and residues at each pole for tanh z.
13. Find the poles and residues at each pole for
3
3( 1) ( 2)( 3)
z
z z z .
14. Evaluate
3cos
(2 3 )c
zdz
i z where C is z = by residue theorem.
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15. Evaluate sinhc
dz
z where C is z = 4 by residue theorem.
16. Evaluate tan
c
zdz where C is z = 2 by residue theorem.
17. Evaluate 2 2
3sin
( / 4)c
zdz
z where C is z = 2 by residue theorem
18. Evaluate
2cos
( 1)( 2)c
z dz
z z
where C is z = 3/2.
19. Evaluate
2 2
2
(cos sin )
( 1) ( 2)c
z z dz
z z
where C is z = 3/2.
20. Evaluate
2
3
( 1)
( 1) ( 2)
z
c
e dz
z z
where C is 1z = 3.
21. Show that
2
2 20
2
sin
d
a b a b
a>b>0 using residue theorem.
22. Show that
2
20 (6 3cos )
d
using residue theorem.
23. Show that
2
20 ( cos )
d
a b
, a>b>0 using residue theorem.
24. Show that
2
0 ( cos )
d
a b
, a>0, b>0 using residue theorem.
25. Show that
22
0, ( 1)
(1 cos )
da
b
using residue theorem.
26. Evaluate by residue theorem 40 1
dx
x
27. Evaluate by residue theorem 2 20 ( 1)
dx
x
28. Evaluate by residue theorem 2 20
cos
( )
xdx
a x
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29. Evaluate by residue theorem 2
sin
( 1)( 4)
xdx
x z
30. Evaluate by residue theorem
4
6( 1)
x dx
x
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UNIT-7 (DESCRIPTIVE QUESTIONS)
1. If f(z) is an analytic function then prove that the transformation W = f(z) is
conformal if ( ) 0f z .
2. Show that the transformation w= c + z is translation where C is a complex
constant.
3. Prove that every bilinear transformation transforms the circles into circles.
4. Prove that every bilinear transformation transforms the straight lines into straight
lines.
5. Prove that every bilinear transformation transforms the inverse points into inverse
points.
6. If 1, 2 3 4, ,z z z z are the four points and 1, 2 3 4, ,w w w w are the corresponding images
under the transformation az b
wcz d
then prove that 1, 2 3 4( , , )z z z z
1, 2 3 4, ,( )w w w w .
7. Find the plot of the image of triangular region with vertices (0, 0), (0, 1) and (1, 0)
under the transformation w = (1-i) z + 3.
8. Find the image of the region when x>0, 0<y<2 under the transformation w = iz +
1.
9. Find the image of the line x = y+1 under the transformation w = i/z.
10. Under the transformation w = 1/z, find the image of the circle 2z i = 2.
11. Find the image of the rectangular region in the z – plane under the
transformation w =ze .
12. Find the plot the rectangular region 0 1,0 2x y under the transformation
w =/ 42 (1 2 )ie z i .
13. Find the image of the region in the z-plane between the lines y=0and y= under
the transformation w= ze .
14. Find the image of the domain in the z plane to the left of the line x= -3 under the
transformation w = 2z
15. Show that the transformation w = 2z maps the circle 1 1z into the
cardioid r = 2(1+cos ) where iw re in the w – plane.
16. Find the image of the line y = 2x under the transformation w =2z .
17. Prove that the transformation w = sin z maps the families of lines x = constant and
y = constant into two families of confocal central conics.
18. Find the image of the infinite strip bounded by x = 0 and x = 4
under the
transformation w = = cos z.
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19. Find the bilinear transformation which maps the points (0, i, 1) into the points (-1,
0, 1).
20. Find the bilinear transformation which maps the points (1, i,-1) into the points (0,
1, ).
21. Find the bilinear transformation which maps the points (1+i,-i, 2-i) into the points
(0, 1, i).
22. Under the transformation w = 1
z i
iz
find the image of the circle 1z in the w –
plane.
23. If w=1
1
iz
iz
find the image of z <1.
24. Find the bilinear transformation which maps the points (-1, 0, 1) into the points
(0, i, 3i).
25. Show that the transformation w =
2 3
4
z
z
transforms the circle
2 2 4 0x y x into the straight line 4u + 3 = 0.
26. Find the image of z <1 and z >1 under the transformation w =1iz
z i
.
27. Find the bilinear transformation which maps the points 1-2i, 2+I, 2+3i into the
points 2+I, 1+3i, 4.
28. Determine the bilinear transformation which maps the points 1+I, 2i, 0 into the
points i, 1,2i.
29. Find the bilinear transformation which maps the points 2, i, 0 into the points 1, 0
and i.
30. Find the image of 1< z <2 under the transformation w = 2iz + 1.
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UNIT- 8 (DESCRIPTIVE QUESTIONS)
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Question Bank
Subject: Mathematics – III Class: B.Tech II Year I Sem Branch : ECE, EEE
Unit – I Special Functions
1. Define Gamma function and evaluate
2
1. (JNTU, Jan. 2003/S)
2. Prove that nnn )1( . (Reduction Formula). (JNTU, Jan. 2003/S)
3. Find (a) )5.3( (b) )5.7(
4. Prove that
02
1
2
1
2
12
thatshowHencedxe x . (JNTU, Jan. 2003/S, Nov. 2003/S,
1999)
5. To show that
0
122
2 dxxen nx Another expression for gamma function.
6. Write
0
12 2
dxex axn in terms of functions.
7. Prove that e dx m my
o
m
1/
8. Show that
0
1
n
nky
k
ndyye .
9. Prove that
1
0
1)1(
!)1()(log
n
nnm
m
ndxxx where n is a positive integer and .1m
10. Prove that
1
0
1
1 )(1log
p
p
q
q
pdy
yy where p > 0, q >0 Hence show that
1
0
1
)(1
log ndyy
n
(JNTU 1996, 1994/S, 2001, 2003)
11. Show that )12....(5.3.12
12
nnn
12. Prove that
0
1 coscos nr
ndxxbxe
n
nax
0
1 sinsin nr
ndxxbxe
n
nax where a,
n are positive and a
bbar 1222 tan,
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13. Show that 2
coscos0
1 m
a
mdxaxx
m
m
(JNTU 1995/S)
14. Prove that
0
1)(log
)1(ax
a
a
adx
a
x
15. Evaluate
0
4 2
7 dxx
16. Show that 3
03
xx e dx
(JNTU, 2003)
17. Show that n = log /11
0
1
y dyn
18. Prove that )2(22
1)(
12nnn
n
19. Show that x e dx
a
m nm ax
m
n
n
0
1
11
(( ) / ) where n and m are positive constants.
Beta Function
20. Define Beta function and prove that Beta function is symmetric. (JNTU, Nov. 2003/S)
21. To prove )(
),(nm
nmnm
(JNTU 1998, 95/S, 94/s, Jan. 2003/S, Nov.
2003/S)
22. To prove
2
0
2
22
2
1
2
1
2
1,
2
1
2
1cossin
qp
qp
qpdxxx qp
23. Prove that sin cos/
2 4
0
25
256
d
24. Show that (m,n) = 2 sin cos/
2 1
2
2 1m
o
n d
and deduce that
sin cos /(( ) / )
(( ) / )
/
n
o
n
o
d dn
n
2
1 21 2
2 2
1/2
25. To prove
2
02
2
2
2
1
sin
n
n
dxxn (JNTU, April, 2003)
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26. Prove that ),(),1( nmnm
mnm
27. Evaluate dxxx
1
0
3 1 using functions.
28. Show that
2
3,42162
2
0
3 dxxx
29. Show that )1,(),1(),( nmnmnm
30. Prove that
1
0
11
0
1
)1()1(),( dx
x
xx
y
dyyqp
qp
qp
qp
q
31. Prove that )1(2
)12(
2
12
n
nn
n
.
32. Legendre’s Duplication Formula: Show that
2
1
2
2)2(
12
nnnn
(JNTU,Jan.2003/S, Nov. 2003/S)
33. Show that
2
04
3
4
1
2
1cot
d (JNTU Nov. 2003/S)
34. Evaluate
2
0
sectan
d (JNTU 1999)
35. Prove that
b
a
nmnm nmabdxxbax )1,1()()()( 1 (JNTU, April. 2003)
36. Express
1
0 1 nx
dx in terms of gamma function (JNTU 1998/S, 2003)
37. Show that
0
1
0
1
)1()1(),( dx
x
xdx
x
xnm
nm
n
nm
m
(JNTU, Jan. 2003/S)
38. Prove that
1
0
1
11
)1(),( dx
x
xxnm
m
nm
(JNTU 2000)
39. Prove that
2
0
42
256
5cossin
d . (JNTU 2003)
40. Prove that
1
0
4
3
4
625
6
5
!31log dx
xx
41. xxxx
sin)(
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42. Given
0
1
sin1
ndx
x
xn
, show that
nnn
sin)1( and hence evaluate
0
41 y
dy
(JNTU 2000/S)
43. Prove that
nnn
sin)1()(
44. Prove that 411
1
04
1
04
2
dx
x
dxdx
x
x (JNTU, Jan. 2003/s)
Bessel’s Differential Equation
45. )()( 1 xJxxJxdx
dnnn
n
(JNTU 2001, April 2003)
46. )()( 1 xJxxJxdx
dn
n
n
n
(JNTU, April 2003)
47. )()()( 1 xJxJx
nxJ nnn
48. )()()( 1 xJxJx
nxJ nnn (JNTU, April 2003)
49. )()()(2 11 xJxJxJ nnn
50. )()()(2
11 xJxJxJx
nnnn
51. Write down the power series expansion for J xn( ) and hence show that
J xx
1 2
2/
( )
sin x.
52. Prove that x
xJxJ
2)()(
2
21
2
21 .
53. Show that )(.cos
sin2
)( 2523 xJevaluateHencex
xx
xxJ
54. Prove that )()( 1 axJaxaxJxdx
dn
n
n
n
55. Show that )()(2
1)( 020 xJxJxJ
56. Prove that )(2
)()( 123 xJx
xJdxxJ . (JNTU, April 2003)
57. Prove that )()( 2
1
2
1 nnnn JJxJxJdx
d.
58. Show that
1
1
111
)(
)()(2)1()1(
qp
qpdxxx qpqp
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59. Prove that J0(x) = 1
0
cos( cos ) x d
60. Evaluate dxxJx )(1
2 .
61. )()1()( xJxJ n
n
n (JNTU, Jan. 2003)
62.
....3sin)(sin)(2)sinsin(
....4cos)(2cos)(2)()sin(cos
31
420
xJxJx
andxJxJxJx
63. If and are the distinct roots of )(axJ n , prove that
ifdxxJxxJ
a
nn 0)()(0
ifxJa
dxxJx
a
nn
2
0
2)(
2
2)(
64. Prove that )()1()( xJxJ n
n
n
65. Prove that ....4cos22cos2)cos(cos 420 JJJx
66. Show that cos (x sin ) = J0 + 2 (J2 cos 2 + J4 cos4+....)
67. ...222sin 531 JJJx
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UNIT - II
Legendre’s Differential Equation:
68. Rodrigue’s Formual: n
n
n
nn xdx
d
nxP )1(
!2
1)( 2 (JNTU 2003/S)
69. Generating function of Legendere :
0
2 )()21( 21
n
n
n zxPzxz (JNTU
2003/S)
70. Prove that )()()1()()12( 11 xnPxPnxxPn nnn
71. Prove that )()()( 1 xPxPxxnP nnn
72. Prove that )()()()12( 11 xPxPxPn nnn
73. Prove that )()()( 11 xnPxPxxP nnn
74. Prove that )()()()1( 1
2 xxPxPnxPx nnn
75. Prove that )()()1()(1 1
2 xPxxPnxPx nnn
76. When n is a +ve integer prove that Pn(x) = 1
12
0
( cos )x x n d
77.
nmn
nmdxxPxP nm
12
2
0)()(
1
1 (JNTU, Jan. 2003/S)
78. Show that n
nn
n
n PthatproveHencexPxP )1()1().()1()(
79. Prove that
1
1
11
2
)32)(12)(12(
)1(2)()(
nnn
nndxxPxPx nn
(JNTU, 2003/S)
80. Show that
1
1
2114
2)()(
n
ndxxPxxP nn
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Objective Type Questions
1. (1) = ……………
2. 1
2
= ……………
3. ( 1)n = ………….. if n is a positive integer.
4. 2
0
xe dx
= ………….
5. ( 1, ) ( , 1)p q p q = ……………….
6. 3
0
xe dx
= ……………………
7. 24
0
7 x dx
= ………………….
8. Value of 0
a
xx dxa
is ………………………..when a > 1.
9. 2
7
0
sin x dx
= …………………….
10. ( ) (1 )n n = ………………….
11. 3 1
4 4
= …………………
12. ,m n in terms of function is ……………………….
13. /2
sinn
o
d
in terms of β function is ………………………….
14. ( )n
n
dx J x
dx
= ………………………
15. 1 1( ) ( )n nJ x J x = …………………………
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16. 2 2
1 2 1 2( ) ( )J x J x = ……………………….
17. 3( )J x dx ………………………
18. ( )nJ x = …………………..
19. sin( sin )x = ………………………..
20. 1
2 2(1 2 )xz z
= …………
21. 2(1 ) ( )nx P x = ………………..
22. ( )nP x = ………………..
23. ( 1)nP = ……………….
24. The value of 1
3 4
1
( ) ( )P x P x dx
is ……………..
25. 1
2
1
( )nP x dx
= …………….
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Unit – III - A Functions of Complex Variable
1. Derive Cauchy’s Riemann Equation in Cartesian co-ordinates.
2. Derive Cauchy Riemann equation in polar coordinates.
3. Prove that the function f(z) = z is not analytic at any point.
4. Determine whether the function 2xy + i(x2- y2) is analytic.
5. Find all values of k, such that f(z)=ex(cos ky+ i sin ky) is analytic.
6. Determine p such that the function f(z) = 2
1 loge(x
2+y2) + i tan –1
y
px be an analytic function
7. Prove that an analytic function with constant real part is constant.
8. Prove that an analytic function with constant imaginary part is constant.
9. Prove that an analytic function with constant modulus is constant.
10. Define analyticity of a complex function at a point P and in a domain D. Prove that the real and
imaginary parts of an analytic function satisfy Cauchy – Riemann Equations.
11. If f(z)=u(xy)+iv(xy) be analytic function, show that u(xy)=c1 and v(xy)=c2 form an orthogonal
system of curves.
12. Show that w = zn (n , a positive integer) is analytic and find it’s derivative.
13. Show that f(z) = 42
2 )(
yx
iyxxy
, z 0
0 , z = 0 is not analytic at z =0 although C –R equations
are satisfied at the origin
14. Find an analytic function f(z) such that Real [f ' (z)] = 3x2 – 4y – 3y2 and f(1+i) = 0.
15. Prove that function f(z) defined by 22
33 )1()1(
yx
iyixzf
, z 0
= 0 , z =0
is continuous and the Cauchy Riemann equations are satisfied at the origin Yet f 1(0)
does not exist.
16. If f(z) = u + iv is an analysis function, show that u and v satisfy Laplaces equation
17. If f(z) is a regular function of Z, prove that 22
2
2
2
2
|)(|4|)(| zfzfyx
.
18. Show that f(x,y) = x3y – xy3 + xy +x +y can be the imaginary part of an analytic function of z =
x+iy.
19. Prove that 22
2
2
2
2
|)(|2|)(Re| ' zfzfalyx
where w =f(z) is analytic.
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20. f(z) is analytic prove that 2222 // yx |f(z)|p=p
2|f'(z)|
2|f(z)|
p-2.
21. Examine the nature of the function f(z) = 104
52 )(
yx
iyxyx
, Z0 f(0) = 0 in the region
including the origin.
22. If w = f(z) is an analytic function of z, prove that
2
2
2
2
yx log )(' zf = 0
23. Find an analytic function f(z) whose real part u = ex (x cosy-y siny).
24. Find the regular function whose real part is u = e2x(x cos 2y – y zsin 2y).
25. Determine the analytic function whose real part is e2x(x cos 2y – y sin 2y).
26. If u(x, y) and v(x,y) are harmonic functions in a region R, prove that the function
x
v
y
u
+i
y
v
x
u is an analytic function.
27. Find the conjugate harmonic of xyeu yx 2cos22 . Hence find f(z) in terms of z.
28. Show that 22 yx
xu
is harmonic.
29. Prove that u=log(x2+y2) is harmonic.
30. Find f(z) =u+iv given that u+v=xy
x
2cos2cosh
2sin
31. Derive the necessary conditions for f(z) to be analytic in Polar-co-ordinates.
32. Find a and b if f(z) = (x2-2xy+ay2)+i (bx2-y2+2xy) is analytic. Hence find f(2) interms of z.
33. For w = exp(z2), find u and v and prove that the curves u(x,y) = c1 and v(x,y) = c2 where c1 and c2
are constants cut orthogonally.
34. Prove that the function f(z) = z is not analytic at any point.
35. If the potential function is loge(x2+y2) find the complex potential function.
36. Show that f (z) = z + 2 z is not analytic anywhere in the complex plane.
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Objective Type Questions
1. If )2( zzw then )(, zfizz ………………………
2. Cauchy –Riemann equations are
…………………………………………………………………………..
3. An analytic function with constant real part is ………………………..
4. An analytic function with constant imaginary part is …………………………..
5. An analytic function with constant modulus is constant is …………………………..
6. The real and imaginary part of an analytic function satisfy
………………………………………..
7. If ),(),()( yxvyxuzf is an analytic function then it represents two family of curves
1),( cyxu and 2),( cyxu forming an ………………………………………………
8. If the derivative exist at all points of z of a region R, then )(zf is said to
be…………………………
9. Functions which satisfy Laplace’s equation in a region R are called ……………………..
in R.
10. A point at which )(zf fails to be analytic is called
……………………………………….of )(zf .
11. The harmonic conjugate of 23 3xyx is …………………………………..
12. The harmonic conjugate of ye x cos is …………………………………….
13. If z
zzfw
1
1)( then
dz
dw……………………………………..
14. The value of k so that 222 kyxx may be harmonic is …………………………
15. If 22 yxu , then the corresponding analytic function is ……………………………
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Unit – III - B Elementary Functions
1. Find the real and imaginary parts of (a) zsin (b) zcos (c) ztan .
2. Find the real and imaginary parts of 2ze .
3. If ,)sin( iyxiBA prove that (JNTU Feb 2008 Set No.4)
(i) 1sinhcosh 2
2
2
2
B
y
B
x
(ii) 1cossin 2
2
2
2
A
y
A
x
4. If ,sincos)sin( ii then prove that 24 sincos . (JNTU Feb 2008 Set No.1)
5. Find the general and the principal values of (i) log e (1+ 3 i) (ii) log e(-1).
6. Find all the roots of .2sin z
7. Find all the roots of the equation .4coshsin z
8. Find the general values of ii ii log, .
9. Determine all values of ii 1)1( .
10. Find the general and principle values of )1log()()log()()31log()( iiiiiiii .
11. If ,)tan( iBAiyx Show that 12cot222 xABA .
12. Find all principle values of )31()31( ii
13. Find all principle values of
)31(
22
3i
i
14. If 1,))tan(log( 22 bawhereibaiyx show that ))tan(log(1
2 22
22yx
ba
a
15. If ivuiCo
4sec prove that )(2)( 22222 vuvu
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Objective Type Questions
1. Real part of coshz is …………………………
2. Imaginary part of sinhz is ……………………….
3. The period of ztan is ……………………….
4. If iyxz then zsin = ………………………………….
5. If iyxz then zsin = …………………
6. If iyxz then zcos = …………………
7. Sin iz = ……………………….
8. Cosh iz = ……………………..
9. Solution set of 0cos z is …………………………………..
10. Solution set of 0sin z is ……………………………………
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Unit – IV Complex Integration
1. Evaluate )(
1
0
2 iyx
i
along y=x2.
2. Evaluate ixyx 2 from A(1,1) to B(2,8) along x=t y=t3.
3. Evaluate dzzi
o2 2 along the imaginary axis to i and o horizontally to 2+i.
4. Evaluate dyyxydxx )3(3( 23)3,1(
)0,0(
2 along the curve y=3x.
5. Evaluate c ( y2 + z2 ) dx +( z2 + x2)dy + ( x2 + y2)dz from ( 0,0,0 ) to ( 1,1,1 ) where C is the curve
x = t, y=t2 , z=t3 in the parameter form.
6. State and prove Cauchy’s integral theorem.
7. Evaluate using Cauchy’s integral formula c
iz
dze3
2
)1( where c: 31 z
8. Evaluate dzzz
z
c
2)1)(32(
712where C is x2+y2=4.
9. Evaluate c
z
zz
dze3)1(
if (i) 0 lies inside C and 1 lies outside C (ii) 1 lies inside C and 0 lies outside
C (iii) both lie inside C.
10. Using Cauchy’s Integral formula, evaluate. dzz
e z
4
2
)1( around the circle 21 z
11. Evaluate dzzz
z
c
)2()1(
1
2 where c is |z-i| = 2.
12. Evaluate dzzz
z
c
)9)(1(
23
2
2
where c is |z-2| = 2
13. Evaluate
C izz
zz2
2 1dz with c : | Z – 1/2 | = 1 using Cauchy’s Integral Formula.
14. Evaluate c
zz
z3
2
)2)(1(
cos where C is |z| =3 by using Cauchy’s integral Formula.
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15. Evaluate
c iz
zz
2
3
)(
12 where C is |z|=2 by using Cauchy’s integral Formula.
16. Evaluate c z
dzz3)1(
log where c : | z - 1| = ( ½ ) , using Cauchy’s Integral Formula
17. Evaluate
C
z
z
dze222
Where c is | Z | = 4.
18. Evaluate
Cdzzz
dz
48Where c is the circle | Z | = 2.
19. Evaluate c Sinhz
dz, where C is the circle | z | = 4
20. Evaluate c
z
az
dzze3)(
where c is any simple closed curve enclosing the point.
z = -a using Cauchy’s integral formula.
21. Evaluate
C
z
iz
z
z
e2
4
3 )( where c: 2z Using Cauchy’s integral theorem.
22. Evaluate
Czz
z
52
42
dz Where C is the circle (i) z = 1 (ii) iz 1 = 2 (iii)
iz 1 = 2
23. Evaluate C
zz
zdz2)2)(1(
Where C is 2z = 2
1
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Objective Type Questions
1. A point at which a function )(zf is not analytic is called
…………………………………………… .
2. If a function )(zf is analytic at all points interior to and on a simple closed curve c.
then
= …………………
3. Let )(zf be an analytic function everywhere on and within a closed contour c. If is any
point within c, then = ……………………..
4. If )(zf is analytic within and on the boundary of a region bounded by two closed curves
and , then = ……………………
5. Let )(zf be an analytic function everywhere on and within a closed contour c. If is any
point within c, then = ……………………..
6. The value of dxz
i
1
0
2 along the line xy is ………………………………
7. The value of
)1,1(
)0,0(
2 2)( dyxydxyx along the line xy is
………………………………
8. The value of
)3,1(
)0,0(
222 )( dyyxdxyx along the line xy 3 is
………………………………
9. The value of the integral c
zz
dz
22 where C: 1z is ……………………………
10. The value of the integral c
zz
dz
22 where C: 12 z is ……………………………
11. dzz
z
c
1
12
2
= ………………………. where C: 1 iz is ……………………………
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12. Value of dzzc
2 wher c is circle 1z is ……………..
13. Value of dzz
zz
c
1
12
where c is circle 2
1z is …………………………
14. Value of dzzz
z
c
52
42
where c is circle 1z is …………………………
15. If c is any simple closed curve and z = a is outside c. then dzaz
c
1=…………….
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Unit – V Complex Power Series
1. State and prove Taylor’s theorem.
2. Expand log(1-2) when |z|<1 using Taylor series.
3. Obtain the Taylor Expansion of ze 1
in the powers of ( z –1 ).
4. Obtain the Taylor series expansion of )1(
)(
zz
ezf
z
about z = 2
5. State and prove Laurent’s Theorem.
6. Obtain all the Laurent series of the function )2)()(1(
27
zzz
z about Z0 = -1
7. Expand the Laurent series of ,)3)(2(
12
zz
z for | z | > 3.
8. Expand )2(
3)(
2
zzz
zzf when (i) |z|<1 (ii) 1<|z|<2 (iii) |z|>2.
9. Find the Laurent series expansion of the function )2)(3)(1(
162
zzz
zz in the region 3<|z+2|<5.
10. Find the Laurent’s expansion of f(z)=)2()1(
27
zzz
z in 1<|z+1|<3.
11. Expand 23
1
2 zz
in 0<|z-1|<1 as Laurent’s series.
12. Obtain Laurent’s expansion for 2)1)(2(
1)(
zzzf
in (i) |z|<2 (ii) |1+z|>1.
13. Show that when | z +1| 1. n
n
znZ 1111
2
14. Show that when | z + 1 | < 1,
1
2 )1)(1(1
n
nznz .
15. Expand 6
1)(
2
zzzf about (i) z = -1 (ii) z = 1.
16. Find the expansion f(z) = )2)(1(
12 zz
When (i) z <1 (ii) 1< z < 2 (iii)
z >2
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Unit – VI The Calculus of Residues
1. Determine the poles of the function and the corresponding residues. )2(
1
2
zz
z
2. Determine the poles of z
zzf
cos)( .
3. Find the residue of 14
2
Z
Zat these Singular points which lie inside the circle | Z | = 2.
4. Find the residue of )1()1(
2)(
22
2
ZZ
ZZzf at each pole.
5. Find the residue of 14
2
Z
Zat these Singular points which lie inside the circle | Z | = 2.
6. Find the residue of 3)1( Z
Ze z
at each of its pole.
7. Determine the poles and residues of f(z)=2
2
)1()2( zzz
z
8. Obtain the Laurent’s expansion of 2)1( z
e z
in the neighborhood of its singular points and hence
find its residue.
9. State and prove Cauchy’s Residue Theorem.
10. (i) Calculate the residue at z=0 of f(z)=2sin2cos
1
z
e z
(ii) Evaluate
0 222 )( ax
dx
11. Show that
2
0cosba
d =
2
0bSina
d=
22
2
ba
, a > b> 0 using residue theorem.
12. Evaluate )()cos(
2
2oba
ba
d
o
13. Evaluate by contour integration
0
21 x
dx.
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14. Evaluate dzzzz
z
c
)2)(1(
34 Where c is the circle | Z | =
2
3
15. Evaluate 52
32
zz
z
c where c in the circle (i)│z│=1 (ii)│z+1-i│=2 (iii)
│z+1+i│=2
16. Evaluate
41 x
dx.
17. Evaluate
c ZZ
Z
52
3
2 dz where C is the circle using residue theorem. (i) | Z | = 1
(ii) | Z+1-i | = 2
18. Evaluate by contour integration
0
21 x
dx.
19. Apply the calculus of residues to prove that dxx
0
22 )1(
1 =
4
20. State and prove Cauchy’s Residue theorem and using it evaluate dzez
c
2/12 where C is
|z|=1.
21. Use the residue theory to show that 2/322
2
0
2 )(
2
)cos( ba
a
ba
d
where a>0, b>0, a>b.
22. Apply the Calculus of residue technique to prove that8
3
)1( 32
x
dx.
23. Prove that
2
0
2
bCosa
dSin =
)(
2 22
2baa
b
, a > b>0.
24. Use method of contour integration to prove that
2
02 21 aCosa
d =
21
2
a
, 0 < a<
1
25. Evaluate
d
0 cos45
cos21
26. Evaluate
d
o
2
cos45
3cos
27. Evaluate
41 22
2
xx
dxx
28. Evaluate dxx
x
16
4
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29. Evaluate
)0(22
aaxx
Sinxdx
o.
30. Evaluate dxx
x
o
22 )1(
cos
31. Using Contour integration evaluate
d
o
2
sin3
cos
32. Evaluate
o ax
dx44
33. Evaluate dxax
mxx
o
22
sin
34. Show that
0221
2
aaCos
Cos =
2
2
1 a
a
, (a2 < 1)
35. Show that
2
02 Cos
d =
3
2
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Objective Type Questions
1. A zero of an analytic function )(zf is that of z for which )(zf = ……………..
2. If then is pole of order …………………………….
3. If )(zf has a simple pole at , then = …………………………..
4. If )(zf has a pole of order n at , then = ……………………………. .
5. Poles of the function are …………………………………..
6. Poles of the function are …………………………………..
7. Poles of the function are …………………………………..
8. Poles of the function are …………………………………..
9. Residue of at is ……………………………………….
10. Residue of at is ……………………………….
11. Residue at of is …………………………………………………..
12. Residue at of is ………………………………..
13. Residue of at is …………………………….
14. Residue of at is …………………………….
15. For a > b > 0, ……………………………..
16. For a > b > 0, ……………………………..
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Unit – VII
Conformal Mapping
1. Define conformal mapping? Let f(z) be analytic function of z in a domain D of the z-plane and let
f’(z)0 in D. Then show that w=f(z) is a conformal mapping at all points of D.
2. State and prove the sufficient condition for w=f(z) to be conformal at the point zo.
3. Find the image of the rectangle R: - < x <; 1/2 < y< 1, under the transformation w = sin z.
4. Find the image of the triangle with vertices at in the z-plane under the
transformation .
5. Discuss the transformation W= sin Z
6. Under the transformation , find the image of the circle .
7. Under the transformation iz
izw
1, find the image of the circle |z|=1 in the w-plane.
8. Find the image of |Z| =2 under the transformation =3z. (ii) Find the points at which = cos
hz is not conformal.
9. Find the image of the triangular region in the z-plane bounded by the lines x=0, y=0 and x+y=1
under the transformation w=2z.
10. Discuss the transformation w=cos hz.
11. Find the images of the following under the transformation w=ez.
(i) the line y=x
(ii) the segment of y-axis given by 0 y
(iii) the left half of the strip 0 y and
(iv) the right half of the strip 0 y .
12. Discuss the transformation zlog
13. Find the bilinear transformations that maps the points 2, i, -2 into 1, i, -1 respectively.
14. Show that the transformation 4
32
z
zw changes the circle x2 + y2 –4x = 0 into the straight
line 4y+3=0.
15. Show that the relation 24
45
z
zw transforms the circle |z| = 1 into a circle of radius unity in
the w-plane.
16. Show that the function w=4/z transforms the straight line x=c in the z-plane into a circle in the
w-plane.
17. Show that the transformation Iz
izw
maps the real axis in the z-plane into the unit circle
|w| =1 in the w-plane.
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18. Find the bilinear transformation which transform the points , i, o in the Z-plane into o, i,
in the w-plane.
19. Find the bilinear transformation which maps the points , i, o in the z-plane into -1, -i, 1 in
the w-plane.
20. Find the bilinear transformation which maps the points into the point
respectively.
21. Find the bilinear transformation that maps the points , into the points .
22. Find the bilinear transformation which maps the points -1, 0, 1 in to the points 0, -1,
respectively.
23. Define bilinear transformation. Find the bilinear transformation that maps 1, i and -1 of the z-
plane onto 0, 1 and of w-plane.
24. Find the transformation which maps the points –1,i,1 of the z- plane onto 1, i,-1 of the w-Plane
respectively.
25. Determine the bilinear transformation that maps the points 1-2i, 2+i, 2+3i into the points 2+i,
1+3i, 4.
26. Discuss the transformation of w=ez.
27. Prove that under the transformation w=1/z the image of the lines y=x-1 and y=0 are the
circle u2+v
2-u-v=0 and the line =0 respectively.
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Objective Type Questions
1. A mapping which preserves the angles between oriented curves in magnitude as well as in
sense is called …………………………………
2. The points which are mapped onto themselves under a conformal mapping are
called……………………..
3. The fixed points of .are …………………………….
4. The fixed points of the mapping are……………………………
5. The Critical points of the mapping are ………………………..
6. Critical points of the mapping are ………………………………
7. The transformation maps a circle onto …………………………….
8. The invariant points of the mapping are …………………………
9. The invariant points of the mapping are………………………
10. Image of under the mapping is ……………………..
11. Bilinear transformation always transforms circles into ………………..
12. Image of under is ……………………..
13. Under the transformation , circle transforms into a ………………….
In w – plane.
14. The Bilinear transformation is a combination of …………………., …………………………, …………………
and ………………………….
15. Bilinear transformation preserves …………………………….. of four points.
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2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 119
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 120
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 121
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 122
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 123
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 124
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 125
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 126
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 127
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 128
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 129
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 130
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 131
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 132
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 133
TIRUMALA ENGINEERING COLLEGE
2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph., Page 134
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