Department of Applied Science & Humanities (Mathematics

Government Engineering College, Nawada

Department of Applied Science & Humanities (Mathematics)

Tutorial Sheet-I Session : 2019-20(Even Sem.) Semester : II

Course/

Branch

: B. Tech./ME Paper Name : Mathematics-II

(102202)

Module : 2 Topic Covered : FIRST ORDER ORDINARY

DIFFERENTIAL EQUATIONS

Name of Faculty: Dr. Rajnesh Kumar

Note: Following are the problems which are required to be done by the students for an overall

understanding of the topics.

1. Find theorder and degree of the following differential equations. State also whether they are linear

or non-linear (i) 1 + dy

dx

2

5/2

=d2y

dx2 (ii) d2y

dx2 + 5dy

dx+ 6y = 0 (iii) y′′ + xyy′ + 3y = 5x (iv)

(y′)2 + 3xy′ + y = 0 (v) y′ = siny

2. Eliminate the arbitrary constants and obtain the following differential equations satisfied by it

(i) y = a cosθx + b sinθx, θ: fixed constant (ii) y = c cos(pt − a), p: fixed constant (iii)

x2 + y2 = a2 (iv) y = 2cx − c2(v) x2 + y2 − 2ay = 0.

3. Reduce to separable form and solve the following differential equations (i) (xy′-y) cos(y/x) +x=0

(ii) xy′ = e−xy − y (iii) dy

dx= ex−y + x2e−y (iv) x2(1 − y)dy + y2(1 + x)dx = 0 (v)

sec2x tany dx + sec2y tanx dy = 0 (vi) y′ = cos(x + y) + sin(x + y) (vii) sin3xdy

dx= siny.

4. Solve the following differential equations (i)dy

dx=

x2y

x3+y3 (ii) (x2 − y2)dx − xy dy = 0 (iii)

(1 + ex/y) + ex/y(1 − x/y)dy = 0 (iv)( x2 + 4y2 + xy)dx − x2dy = 0 (v) (3xy + y2) dx +

(x2 + xy)dy = 0.

5. Solve the following differential equations(i)dy

dx=

x−y−2

2x−2y−3 (ii)

dy

dx=

2x−6y+7

x−3y+4 (iii)

dy

dx=

x+2y−3

2x+y−3 .

6. Solve the following differential equations

(i)(x2 + 1)dy

dx+ 2xy = 4x2 (ii) x

dy

dx+ y = y2 log x(iii) x

dy

dx= 2y + x4 + 6x2 + 2x (iv) (1 +

y2)dx = (tan−1y − x)dy (v) (x + 1)dy

dx− y = e3x(x + 1)2(vi) (1 + x2)

dy

dx+ y = etan −1x

7. Solve the following differential equations(i) (y2exy 2+ 4x3)dx + (2xyexy2

− 3y2)dy = 0

(ii)(2x3 + 3y2x − 7x)dx + (3x2y + 2y3 − 8y)dy = 0 (iii)(x2y − 2xy2)dx − (x3 − 3x2y)dy =

0 (iv) (1 − xy)ydx − x(1 + xy)dy = 0 (v)(xy3 + y)dx + 2(x2y2 + x + y4)dy = 0 (vi) y(xy +

2x2y2)dx + x(xy − x2y2)dy = 0 (vii) (cosy + ycosx)dx + (sinx − x siny)dy = 0.

8. Solve: (i) xdy

dx+ y = x3y6 (ii) y′ + 4xy + xy3 = 0 (iii) y′ − y = y2(sinx + cosx).

9. Solve : (i) dy

dx

2− 5

dy

dx+ 6 = 0 (ii)

dy

dx

2+ 2x

dy

dx− 3x2 = 0 (iii) x4

dy

dx

2− x

dy

dx− y = 0.

10. Solve: (i) y = 2px + y2p3 (ii) y = xy′ + (y′)2 (iii) y = xy′ − e2y′ (iv) y = xy′ −1

y′ .

11. Find the orthogonal trajectories of the hyperbolas x2 − y2 = c.

12. Find the orthogonal trajectories of the family of circles passing through the points (0, 2) and (0, -

2).

13. Find the orthogonal trajectories of the following family of curves (i)r = c(1 + cosθ) (ii) r2 =

c sin(2θ).

14. A body is heated to 100℃ and placed in air at 20℃ . After one hour its temperature is 60℃. How

much additional time is required for it to cool to 30℃.

15. In a radioactive decay, initially 50 mg of the material is present and after two hours, the material

has lost 10% of its original mass. Find the mass at any time t and the half-life of the material.

16. A particle falls down from rest in the air whose resistance is prepositional to the square of the

velocity. Find the velocity as a function of x.

Text / Reference Books:

1. Peter V. O` Neil, A text book of Engineering Mathematics, Thomson (Cengage Learning), 2nd

Edition, 2010.

2. B.S.Grewal, Advanced Engineering Mathematics, Khanna Publishars, 40th

Edition, 2010.

3. E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley and Sons, New York, 2005.

4. B.V. Ramanna, “Higher Engineering Mathematics”, Tata Mcgraw Hill Publishing Company Ltd.,

2008.

5. R.K. Jain and S.R.K. Iyengar, “Advanced Engineering Mathematics”, Narosa Publishing House,

2008.



Tutorial Sheet-II Session : 2019-20(Even Sem.) Semester : II

Course/

Branch

: B. Tech./ME Paper Name : Mathematics-II

(102202)

Module : 3 Topic Covered : ORDINARY DIFFERENTIAL

EQUATIONS OF HIGHER

ORDERS Name of Faculty: Dr. Rajnesh Kumar



1. Examine whether the following functions are linearly independent (i) 1, 𝑐𝑜𝑠𝑥, 𝑠𝑖𝑛𝑥 (ii)

𝑙𝑛𝑥, 𝑙𝑛 𝑥2, 𝑙𝑛 𝑥3 (iii) 𝑒−𝑥 , 𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥 (iv) 𝑥, 𝑥2 , 𝑥3 (v) 𝑒𝑥 , 𝑒2𝑥 , 𝑒3𝑥 .

2. Find a general solution of the following differential equations: (i) 𝑦′′ − 4𝑦 = 0 (ii) 𝑦′′ − 𝑦′ −

2𝑦 = 0 (iii) 𝑦′′ + 𝑦′ − 2𝑦 = 0 (iv) 𝑦′′ − 4𝑦′ − 12𝑦 = 0 (v) 𝑦′′ + 9𝑦′ = 0 (vi) 9𝑦′′ − 12𝑦′ +

4𝑦 = 0 (vii) 𝑦′′ − 𝑦′ − 6𝑦 = 0 (viii) 4𝑦′′ + 4𝑦′ + 𝑦 = 0 (ix) 𝑦′′′ − 𝑦′′ − 5𝑦′ + 6𝑦 = 0 (x)

8𝑦′′′ − 12𝑦′′ + 6𝑦′ − 𝑦 = 0 (xi) 𝑦𝑖𝑣 − 𝑎2𝑦 = 0 (xii) 𝑦𝑖𝑣 + 32𝑦′′ + 256𝑦 = 0.

3. Solve the following differential equations: (i) 𝑦′′′ + 𝑦 = 𝑒𝑥 + 2𝑒−𝑥 (ii) 𝑦′′ − 4𝑦′ + 3𝑦 =

𝑠𝑖𝑛3𝑥 𝑐𝑜𝑠2𝑥(iii) 𝑦′′ − 4𝑦′ + 4𝑦 = 𝑒𝑥 + 𝑠𝑖𝑛2𝑥 (iv) 𝑦′′ + 4𝑦 = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠3𝑥 (v) 𝑦′′ − 4𝑦 = 𝑥2

(vi) 𝑦′′ − 2𝑦′ + 3𝑦 = 𝑐𝑜𝑠𝑥 + 𝑥2 (vii) 𝑦′′ − 4𝑦′ + 4𝑦 = 𝑥2 + 𝑒𝑥 + 𝑐𝑜𝑠2𝑥 (viii) 𝑦′′ − 2𝑦′ + 𝑦 =

𝑥𝑒𝑥𝑠𝑖𝑛𝑥 (ix) 𝑦′′ − 3𝑦′ + 2𝑦 = 𝑠𝑖𝑛2𝑥 + 𝑥𝑒𝑥 .

4. Solve by method of variation of parameters: (i) 𝑦′′ + 𝑎2𝑦 = 𝑠𝑒𝑐 𝑎𝑥 (ii) 𝑥2𝑦′′ + 𝑥𝑦′ − 𝑦 = 𝑥2𝑒𝑥

(iii) 𝑥2𝑦′′ − 4𝑥𝑦′ + 6𝑦 = 𝑠𝑖𝑛(𝑙𝑜𝑔 𝑥) (iv) 𝑦′′ + 4𝑦 = 𝑐𝑜𝑠 𝑥 (v) 𝑦′′ + 𝑎2𝑦 = 𝑐𝑜𝑠𝑒𝑐ax (vi)

𝑦′′ + 𝑦 = 𝑡𝑎𝑛 𝑥 (vii) 𝑦′′ + 6𝑦′ + 9𝑦 =𝑒−3𝑥

𝑥 (viii) 𝑦′′ + 4𝑦′ + 4𝑦 = 𝑒−2𝑥𝑠𝑖𝑛 𝑥 (ix) 𝑦′′ − 𝑦 =

2

1+𝑒𝑥

(x) 𝑥2𝑦′′ + 𝑥𝑦′ − 𝑦 = 𝑥2𝑦 (xi) 𝑥2𝑦′′ + 3𝑥𝑦′ + 𝑦 =1

(1−𝑥)2.

5. Solve the following differential equations:(i) 𝑥𝑦′′ − (2𝑥 − 1)𝑦′ + (𝑥 − 1)𝑦 = 0 (ii) (1 −

𝑥2)𝑦′′ + 𝑥𝑦′ − 𝑦 = 𝑥 (1 − 𝑥2 )3/2(iii) 𝑦′′ − 𝑐𝑜𝑡 𝑥 𝑦′ − (1 − 𝑐𝑜𝑡 𝑥) 𝑦 = 𝑒𝑥𝑠𝑖𝑛 𝑥 .

6. Solve: (i) 𝑦′′ − 2𝑡𝑎𝑛 𝑥 𝑦′ + 𝑦 = 0 (ii) 𝑦′′ − 4𝑥𝑦′ + (4𝑥2 − 1)𝑦 = −3𝑒𝑥2𝑠𝑖𝑛 2𝑥 (iii) 𝑥2𝑦′′ −

2(𝑥 + 𝑥2)𝑦′ + (𝑥2 + 2𝑥 + 2)𝑦 = 0 (iv) (𝑐𝑜𝑠 𝑥) 𝑦′′ + (𝑠𝑖𝑛 𝑥) 𝑦′ − 2(𝑐𝑜𝑠3𝑥)𝑦 = 2𝑐𝑜𝑠5𝑥 .

7. Solve: (i) 𝑥3𝑦′′′ + 2𝑥2𝑦′′ + 2𝑦 = 𝑥 +1

𝑥 (ii) 𝑥2𝑦′′ − 5𝑥𝑦′ + 3𝑦 = 𝑙𝑛 𝑥 (iii) (𝑥 + 1)2𝑦′′ + (𝑥 +

1)𝑦′ + 𝑦 = 4𝑐𝑜𝑠(𝑙𝑜𝑔(1 + 𝑥)) (iv) (2𝑥 + 5)2𝑦′′ + 6(2𝑥 + 5)𝑦′ + 8𝑦 = 𝑥 .

8. Find the regular and singular points of the differential equations (i) (1 − 𝑥2)𝑦′′ − 2𝑥𝑦′ + 𝑛(𝑛 +

1)𝑦 = 0 (ii) 𝑥2𝑦′′ + 𝑎𝑥𝑦′ + 𝑏𝑦 = 0.

9. Classify the singular points of the following equations (i) 𝑥2𝑦′′ + (𝑠𝑖𝑛 𝑥)𝑦′ + (𝑐𝑜𝑠 𝑥)𝑦 = 0 (ii)

𝑥3(𝑥2 − 1)𝑦′′ − 𝑥(𝑥 + 1)𝑦′ − (𝑥 − 1)𝑦 = 0 (iii) 𝑥2𝑦′′ + 2𝑥𝑦′ + (𝑥2 − 𝑛2)𝑦 = 0.

10. Find the power series solution about 𝑥 = 0, of the differential equation (i) 𝑦′′ − 2𝑦 = 0 (ii)

(1 − 𝑥2)𝑦′′ − 2𝑥𝑦′ + 2𝑦 = 0.

11. Find the power series solution about 𝑥 = 2 of the equation 𝑦′′ + (𝑥 − 1)𝑦′ + 𝑦 = 0.

12. Find the series solutions of the following differential equations by the Frobenious method: (i) )

𝑥2𝑦′′ + 2𝑥𝑦′ + (𝑥2 − 𝑛2)𝑦 = 0 (ii) 9𝑥(1 + 𝑥)𝑦′′ − 6𝑦′ + 2𝑦 = 0 (iii) (1 − 𝑥2)𝑦′′ − 2𝑥𝑦′ +

6𝑦 = 0.

13. Find the series solutions about the indicated point of the following differential equations by the

Frobenious method: (i) 2(1 − 𝑥)𝑦′′ − 𝑥𝑦′ + 𝑦 = 0, x=1 (ii) 𝑥(𝑥 − 2)𝑦′′ + 4𝑦′ + 3𝑦 = 0, 𝑥 = 2.

14. Express 𝑃(𝑥) = 3𝑃3(𝑥) + 2𝑃2(𝑥) + 4𝑃1(𝑥) + 5𝑃0(𝑥) as polynomial in 𝑥, where 𝑃𝑚 (𝑥) is the

Legendre polynomial of order 𝑚.

15. Express 𝑓(𝑥) = 𝑥4 + 2𝑥3 − 6𝑥2 + 5𝑥 − 3 in the terms of Legendre polynomials.

16. Show that (i) 𝑃𝑛(1) = 1 (ii) 𝑃𝑛(−𝑥) = (−1)𝑛 ) 𝑃𝑛(𝑥) (iii) 𝑃′𝑛(1) = 𝑛(𝑛 + 1)/2 (iv)

𝑃𝑛(𝑥) 𝑑𝑥 = 01

−1 (v) ) 𝑃𝑛

2(𝑥) 𝑑𝑥 =2

2𝑛+1

1

−1.

17. Show that 𝐽𝑛(𝑥) is a even function for n even and an odd function for 𝑛 odd where n is an integer.

18. Prove that (i) 𝐽1/2(𝑥) = 2

𝜋𝑥𝑠𝑖𝑛 𝑥 (ii) 𝐽−1/2(𝑥) =

2

𝜋𝑥𝑐𝑜𝑠 𝑥 (iii) 𝐽3(𝑥) =

8

𝑥2 − 1 𝐽1(𝑥) −

4

𝑥𝐽0(𝑥) (iv) 𝐽5/2(𝑥) =

2

𝜋𝑥

1

𝑥2 (3 − 𝑥2)𝑠𝑖𝑛 𝑥 −3

𝑥𝑐𝑜𝑠 𝑥 (v) 𝐽−5/2(𝑥) =

2

𝜋𝑥

1

𝑥2 (3 − 𝑥2)𝑐𝑜𝑠 𝑥 +

3

𝑥𝑠𝑖𝑛 𝑥 .

Text / Reference Books:

1. Peter V. O` Neil, A text book of Engineering Mathematics, Thomson (Cengage Learning), 2nd

Edition, 2010.

2. B.S.Grewal, Advanced Engineering Mathematics, Khanna Publishars, 40th

Edition, 2010.

3. E. Kreyszig, “Advanced Engineering Mathematics”, John Wiley and Sons, New York, 2005.

4. B.V. Ramanna, “Higher Engineering Mathematics”, Tata Mcgraw Hill Publishing Company Ltd.,

2008.

5. R.K. Jain and S.R.K. Iyengar, “Advanced Engineering Mathematics”, Narosa Publishing House,

2008.



Tutorial Sheet-III Session : 2019-20(Even Sem.) Semester : II

Course/

Branch

: B. Tech./ ME Paper Name : Mathematics-II

(102202)

Module : 4 Topic Covered : Complex Variables-

Differentiation




1. Check whether the following limits exist or not

(i)z

zz

z

)Im()Re(lim

0→, (ii)

0limz

z

z→, (iii)

2

20

Imlimz

z

z→.

2. Find out whether the following functions are continuous at origin or not, where (0) 0f = ,

(i) 2

2

Re z

z, (ii)

2Im z

z, (iii)

2 1Rez

z

.

3. Examine the differentiability of the following complex valued functions

(i) ( )f z z= at origin, (ii) 2

( )f z z= at 0z = , (iii) 2( )f z z= .

4. Determine the analyticity of the following functions.

(i) zzf sin)( = , (ii) )(2)( 22 yxixyzf ++= , (iii) 2( ) (cosh sinh )xf z e y i y= + .

5. Show that the function ,)(4−−= zezf z 0 and 0)0( =f is not analytic at the origin even through

CR equation are satisfied.

6. Examine which of the following functions are harmonic:

(i) 23 34),( xyxyxyxu −−= , (ii) )sincos(),( yyyxeyxu x += − .

7. Using the C-R equations find the harmonic conjugate of the following functions

(i) yxxyyxu 32222 +−−−= , (ii) )log(2

1),( 22 yxyxu +=

where, ( )f z u iv= + is an analytic function.

8. Find the analytic function f(z) using Milne –Thomson method:

(i) yxu coshcos= (ii) ).4)((),( 22 yxyxyxyxu ++−=

9. Show that the function ivuzf +=)( , where

=

+

+

=

0,0

0,)(

)( 104

52

z

zyx

iyxyx

zf

Satisfies the Cauchy-Riemann equations at 0=z . Is the function analytic at 0=z ?

10. Deterrmine the analytic function ( )f z u iv= + , if )cosh(cos2

sincos

yx

exxvu

y

−

−+=−

−

and .0)2( =

f

11. Prove that an analytic function with constant modulus is constant.

12. Find the bilinear transformation which maps the points 2zi,z2,z 321 −=== into the points

1wi,w1,w 321 −=== .

13. Discuss the application of the transformation iz

1izw

+

+= to the areas in the z-plane which are

respectively inside and outside the unit circle with its centre at the origin. 14. Find the general homographic transformation which leaves the unit circle invariant. 15. Every bilinear transformation maps circles or straight lines into circles or straight lines.

16. Show that the transformation ,tanh 1 zw −= maps the upper half of the z -plane conformally on the

strip .2

10 v

17. Show that the resultant (or product) of two bilinear transformations is a bilinear transformation. Explain the concept of isogonal mapping and conformal mapping with illustrative examples.

18. Find the fixed points and the normal of the following bilinear transformation: (i)2−

=z

zw (ii)

.1

1

+

−=

z

zw

19. Show that the line x3y = is map onto the circle under the bilinear transformation iz

izw

+

+=

4

2. Find

the centre and radius of image circle. 20. Find the bilinear transformation that maps the point 0zi,z,z 321 === into the points

=== 321 wi,w0,w .

21. Show that the inverse of the point a with respect to the circle R=c-z is the point )(

Rc

2

ca −+ .

Text / Reference Books: 1. R. K. Jain & S. R. K. Iyengar. “Advanced Engineering Mathematics,” Narosa Publishing House Pvt.

Ltd., 3 Ed., 2011

2. J. K. Goyal & K. P. Gupta. “Functions of a Complex Variable,” Pragati Prakashan., 17th edition 2003

3. Shanti Narayan & P.K. Mittal. “ Theory of Functions of a Complex Variable,” S. Chand Revised Ed,

2010

4. Ruel V. Churchill, Complex variables and Applications

5. S. Ponnusamy. “Foundation of Complex Analysis,” Narosa Publishing House Pvt. Ltd., 2nd Ed. 2005

6. M. R. Spiegel. “Theory of Complex Variable,” Mc - Graw HiIl Publication, 1981.

7. Murray Spiegel, Seymour Lipschutz, John Schiller & Dennis Spellman. “ Schaum's Outline of

Complex Variables, 2ed”

http://www.mhprofessional.com/contributor.php?id=46331






Tutorial Sheet-IV Session : 2019-20(Even Sem.) Semester : II

Course/

Branch

: B. Tech./ ME Paper Name : Mathematics-II

(102202)

Module : 5 Topic Covered : Complex Variables-

Integration




1. If a function )(zf is analytic for finite values of z , and is bounded, then )(zf is constant.

2. Find upper bound of integral −

+

C

iz

dzz

zLogez

2

)3(2

3

, 3/0,2: == iezzC .

3. What curve is represented by the function 36)2(9)1(4 22 =++− yx .

4. Find the value of the integral dzixyx

i

+

+−

21

0

32 )3( along the real axis from 0=z to 1=z and then, along

a line parallel to the imaginary axis from 1=z to iz 21+= .

5. Use ML-inequality to show that 2

1

12

+C

dzz

zwhere C is the straight line segment from 2=z to

iz += 2 .

6. Find the value of the integral izdzzz

e

C

z

4,)2)(1(

2

=−− .

7. Find the value of the integral dyxy

ydx

yx

x

C

2222 −+

− where C is the boundary of the triangle with the

vertices (2, 0), (4, 0) and (4, 3).

8. State and prove fundamental theorem of integral calculus.

9. Evaluate the following integrals by Cauchy’s integral formula

a. dzz

zzc −

++

2

652

where c is the circle 13 == zandz ,

b. −

cdz

z

z

2

2

)6

(

sin

where c is the circle 1=z ,

c. −c

dzz 1

1

2 where c is the circle 2=z .

10. If the curve 20),exp()( = titt then find the integral +

+

dzz

z

)1(

12

.

11. If Cwzwfzfwzf +=+ ,)()()( and )(zf is analytic in C , show that czzf =)( , where c is

constant.

12. Prove that non constant entire function is unbounded.

13. Suppose )(zf is entire function such that 2

)( zzf , ifif == )2(,2)1( find )(zf .

14. Find the radius of convergence of the following power series (i)

++

n

1iz

i 2(ii)

n

nz

2

n i2

+.

15. Find the radius of convergence of the following power series:

a.

= +=

0 14)(

nn

nzzf

b. .!

)(

0

=

=

nn

n

n

znzf

16. Find the region of convergence of the series (i)

=1n

n

n log

z (ii) ( )

( )

=

−−

−−

1

121

!121

n

nn

n

z.

17. Find the first four terms of the Taylor’s series expansion of the function

)(tan)( 1 zzf −= about 0=z .

18. Obtain the Taylor and Laurent’s series which represents the function 3)2)(z(z

1z2

++

− in the regions (i)

2z (ii) 32 z (iii) 3z .

19. Expand )2)(1(

23)(

++

−=

zzz

zzf in Laurent’s series valid for region .3|2|1 + z

20. Expand )9(

1897)(

3

2

zz

zzzf

−

−+= in the region (i) 3||0 z (ii) 3z (iii) 21 z .

21. Find the kind of singularity of the function

−=

z-1

1sin

2zf(z)

2z.

22. What kind of singularity has the function

(i)

=

z

1cos

1f(z)

at 0=z (ii) ( )1/z cosec

at =z .

23. Discuss the singularity of the following functions:

a. z

ezf

z/1

)( =

b. .11

1cos)( =

−= zat

zzf

24. Find the poles and residue of the poles of the following function:

)3)(4)(2(

)(4

++−=

zzz

zzf

25. Determine the nature of the pole at the origin of the function mzz sin

ef(z)

z

= .

26. Show that the function 21/ze− has no singularities.

27. Find the residue of zcot(z) = at the points n=nz for 1,2,...n = what is the nature of singularity at

=z ? Justify your answer.

28. If an analytic function f(z) has a pole of order m at a=z , then 1/f(z)has a zero of order m at a=z

and conversely.

29. If a=z is an isolated singularity of f(z) and if f(z) is bounded on some deleted neighborhood of a ,

then prove that a is a removable singularity.

30. Find residues of (i)2zz

12z2 −−

+ (ii)

12

3

−z

z at .=z

31. Evaluate (i) +

2π

0

2

cosba

dsin

(ii) dx

+0

22

2

)1(x

x.

32. By contour integration, prove that 4x

xsin

0

2 =

dx

.

33. Evaluate the following integrals using Cauchy Residue Theorem:

a.

2:;)12()1(

75

2=

+−

− zCdz

zz

z

c

b. .2:;)4)(1(

12

2

2

=+−

+

C

zCdzzz

z

34. Evaluate the following integrals by contour integration:

a. +

2

0 cos45

d

b. .cos45

2cos2

0

−

d

35. Apply calculus of residue to prove that:

−=

+−

2

0

2

2

2,

1

2

cos21

2cos

a

a

aa

d (a2 < 1 ).

36. Evaluate the following integrals by contour integration:

a.

− + )1( 2x

dx

b.

+022

.)(

sindx

ax

xx

37. Evaluate the following integrals using Cauchy Residue Theorem:

=−−

+−

C

zCdzzz

zz1:;

)13()12(

523302

2

.

38. Apply calculus of residue to prove that:

.0);1(2)(

sin

222−=

+

− ae

adx

axx

mx ma

C

39. Verify the integral ,sin1x

x

0

-a

adx =

+

10 a .

40. Let f(z) be analytic in Rz

and Mf(z)

onRz =

. Then prove that .R

M(0)'f

Text / Reference Books: 1. R. K. Jain & S. R. K. Iyengar. “Advanced Engineering Mathematics,” Narosa Publishing House Pvt.

Ltd., 3 Ed., 2011

2. J. K. Goyal & K. P. Gupta. “Functions of a Complex Variable,” Pragati Prakashan., 17th edition 2003

3. Shanti Narayan & P.K. Mittal. “ Theory of Functions of a Complex Variable,” S. Chand Revised Ed,

2010

4. Ruel V. Churchill, Complex variables and Applications

5. S. Ponnusamy. “Foundation of Complex Analysis,” Narosa Publishing House Pvt. Ltd., 2nd Ed. 2005

6. M. R. Spiegel. “Theory of Complex Variable,” Mc - Graw HiIl Publication, 1981.

7. Murray Spiegel, Seymour Lipschutz, John Schiller & Dennis Spellman. “ Schaum's Outline of

Complex Variables, 2ed”





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Department of Applied Science & Humanities (Mathematics