m III Course File 2011 12

TIRUMALA ENGINEERING COLLEGE

2011-M-III Prof. P.Shantan kumar…M.Sc.,M.Phil.,B.Ed.,D.Ph.,

COURSE FILE




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



CONTENTS

ACADEMIC CALENDER

SYLLABUS

TEACHING SCHEDULE

LESSON PLAN

LECTURE NOTES

ASSIGNMENTS(UNIT WISE)

IMPORTANT QUESTIONS (UNIT WISE)

JNTU PREVIOUS YEARS QUESTION PAPERS



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.



SYLLABUS



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)



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.



TEACHING SCHEDULE




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,


3

----

SHANKARAIAH

Properties-Rodrigue’s

formula, Orthogonality.

3

----

SHANKARAIAH

Chebycher’s polynomials,


3

----

SHANKARAIAH

Properties, Orthogonality.

2

---- SHANKARAIAH



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



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

----



LESSON PLAN




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,


1

11

20-07-11

Properties,

Orthogonality.

1



12

21-07-11



1

13

25-07-11



1

14

26-07-11



1

15

27-07-11


formula,

Orthogonality.

1

16

28-07-11


formula,

Orthogonality.

1

17

01-08-11


formula,

Orthogonality.

1

18

02-08-11

Chebycher’s

polynomials,


1

19

03-08-11

Chebycher’s

polynomials,


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-


1



24

11-08-11

Continuity-

Differentiability-


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


General power z6,

1

30

23-08-11


General power z6,

1

31

24-08-11

Principle value,


1

32

25-08-11

Principle value,


1

33

26-08-11

Line integral-

Evaluation along a

path and by indefinite

integration

1



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-


series-.

1

41

13-09-11

Radius of

convergence-


series-.

1

42

14-09-11


Laurent series

1

43

15-09-11


Laurent series

1

44

19-09-11

singular point-Isolated

singular point

1

45

20-09-11

Pole of order m-

Essential singularity

1



46

21-09-11

Pole of order m-

Essential singularity

1

47

22-09-11


Residue by formula

and by Laurent series

1

48

26-09-11


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



57

11-10-11

Determination of


mapping 3 given

points

1

58

12-10-11

Determination of


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,


Planner graphs

1

64

24-10-11

Regular,


Planner graphs

1

65

25-10-11

Regular,


Planner graphs

1

66

26-10-11

Hamiltonian and

Eulerian Circuits

1

67

27-10-11

Hamiltonian and

Eulerian Circuits

1



68

28-10-11


Minimum Spanning

Ttree.

1

69

29-10-11


Minimum Spanning

Ttree.

1

70

Signature of the faculty Signature of the H.O.D.



ASSIGNMENTS(UNIT WISE)




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





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





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





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)





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





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



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





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





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)





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).





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.





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)



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.





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)



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.



IMPORTANT QUESTIONS

(UNIT WISE)



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



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



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]



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



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



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



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.



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.



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)



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).



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.



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.



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



29. Evaluate by residue theorem 2

sin

( 1)( 4)

xdx

x z

30. Evaluate by residue theorem

4

6( 1)

x dx

x



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.



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.



UNIT- 8 (DESCRIPTIVE QUESTIONS)



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,



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)



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)(



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



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



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



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 = …………………………



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

= …………….



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.



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.




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 ……………………………



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




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 ……………………………………



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.



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




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 ……………………………



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=…………….



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



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.



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



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




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 …………………………………..




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, ……………………………..



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.



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.




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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