List of Practice questions on Mathematics III Unit I: Ordinary Differential Equations Sub Topic: Difference Equations Q1. Form difference equation by eliminating the arbitrary constants 𝑎 and 𝑏 from the relations (i) 𝑦𝑛 = 𝑎 cos 𝑛𝜃 + 𝑏 sin 𝑛𝜃 (ii) 𝑦𝑛 = 𝑎𝑛2 + 𝑏𝑛. Q2. Form difference equation from the relation gives as

log(1+𝑧) (1+𝑧)

= 𝑦0 + 𝑦1 𝑧 + 𝑦2 𝑧 2 + … … … +

𝑦𝑛 𝑧 𝑛 . Q3. Solve 𝑦𝑛+1 − 2 cos 𝛼 𝑦𝑛 + 𝑦𝑛−1 = 0. Q4. Solve the difference equation 𝑦𝑚+3 + 16𝑦𝑚−1 = 0. Q5. Solve the difference equation 𝑢𝑛+3 − 2𝑢𝑛+2 − 5𝑢𝑛+1 + 6𝑢𝑛 = 0. Given that 𝑢0 = 1, 𝑢1 = 2, 𝑢2 = 3. Q6. A series of values of 𝑦𝑛 satisfies the relation 𝑦𝑛+2 + 𝑎𝑦𝑛+1 + 𝑏𝑦𝑛 = 0. Given that 𝑦0 = 0, 𝑦1 = 1, 𝑦2 = 𝑦3 = 2; show that 𝑦𝑛 = 2(

𝑛⁄ ) 𝑛𝜋 2 sin . 4

Q7. The integers 0, 1, 1, 2, 3, 5, 8, 13, 21, … … … 𝑛 are said to form a Fibonacci sequence. Form the Fibonacci difference equation and solve it. Q8 . Show that 𝑛 straight lines, no two of which are parallel and no three of which meet in a 1

point, divide a plane into 2 (𝑛2 + 𝑛 + 2) parts. Yet 𝑦𝑛 denote the number of sub regions formed by 𝑛 straight lines. Q9. A plant is such that each of its seeds when one year old produces 8-fold and produces 18fold when two years or more. A seed is planted and as soon as a new seed is produced it is planted. Taking 𝑦𝑛 to be the number of seeds produced at the end of the nth year, show that 𝑦𝑛−1 = 8𝑦𝑛 + 18(𝑦1 + 𝑦2 + … … … + 𝑦𝑛−1 ). Hence show that 𝑦𝑛+2 − 9𝑦𝑛−1 − 10𝑦𝑛 = 0 and solve it. Q10. A sequence of numbers is such that the nth number of the sequence is the sum of twice the (𝑛 − 1)th and three times the (𝑛 − 2)th numbers, where 𝑛 ≥ 2. The first number is zero and second is unity. Find the nth number of sequence. Q11. Solve the difference equation ∆2 𝑢𝑥 + 2∆𝑢𝑥 + 𝑢𝑥 = 2𝑥 . 1

𝑛

Q12. Solve 𝑦𝑛+2 = (2 cos 2) 𝑦𝑛+1 − 𝑦𝑛 + sin 2 .

Q13. Solve 𝑢𝑘+2 + 𝑎2 𝑢𝑘 = cos 𝑎𝑘. Q14. Solve the difference equation 𝑢𝑛+2 − 4𝑢𝑛+1 + 4𝑢𝑛 = 𝑛2 2𝑛 . Q15. A beam of length 𝑙, supported at 𝑛 points carries a uniform load 𝑤 per unit length. The bending moments 𝑀1 , 𝑀2 , 𝑀3 , ………𝑀𝑛 at the supports satisfy the Clapeyron’s equation: 1

𝑀𝑟+2 + 4𝑀𝑟+1 + 𝑀𝑟 = − 2 𝑤𝑙 2 . If the beam weighing 30 kg is supported at the ends and at two other supports dividing beam into three equal parts of 1 meter length (as given in Fig. 1). Show that the bending moments at two middle supports are 1 and 16 units, respectively.

Figure: 1

Q16. Solve the difference Equation: y h3  5 y h 2  8 y h1  4 y h  h.2 h

Q17. Solve the difference Equation: u x  2  2mu x 1  (m 2  n 2 )u x  m x

Q18. Solve y k 2  6 y k 1  8 y k  3k 2  2  5.3k Q19. Solve E  ayk  cosnk Apply Generating function technique to solve Q20-Q22: Q20. y(t  1)  3 y(t )  t , Q21. Solve yk 2  2 yk 1  yk  1

y0  1 y0  1 ,

y1  1 .

y0  1 ,

Q22. Solve yk 2  5 yk 1  6 yk  2

y1  1 .

Q23. Express the second order difference equation xt 1  3xt  2xt 1  0 as a system of two first order difference equations and hence solve it with initial values x0  0 , x1  1 by using matrix method.

Solve Q9-Q11 by matrix method z k 1  yk  2z k

Q24. yk 1  2 yk  z k ,

Q25. xt 1  4xt  3xt 1  0 with Q26. y k 2  y k  sin  k  1 2

y1  1 , z1  0

x0  1 , x1  1 y0  1 , y1  1



Q27. Show that the system of three difference equations can be written in the form Vn1  AVn  xn  Where Vn   y n   z n 

n=0, 1, 2  3 5 2 A  1  1 1 2 1 3

Show further that the system of equation can be written as Vn1  A nV0

n=0, 1, 2

Sub Topic: Legendre and Bessel ordinary differential equations 1. Show that x  0 is not an ordinary point of y  x 2 y  x y  0. 2. Investigate the nature of the point x  0 for the differential equation x4 y  ( x2 sin x) y  (1  cos x) y  0 3. Find the first three nonzero terms of each of two linearly independent Frobenius series solution of 2x2 y  (sin x) y  (cos x) y  0 . 4. Investigate the nature of the point x  0 for the differential equation x3 y  x( x  sin x) y  (1  cos x) y  0 5. Find the power series solution of x2 y  6(sin x) y  6 y  0 . 6. Find the power series solution of differential equation 2x2 y  (6x  x2 ) y  xy  0 . 7. Apply the method Frobenius series solution of Bessel’s equation of order ½, x2 y  xy  ( x2 1/ 4) y  0, derive the it’s general solution for x  0 ,

cos x sin x .  c1 x x 8. Find the general solution of the x2 y  xy  ( 2 x2  n2 ) y  0 . y( x)  c0

9. Evaluate (i)  x 2 J 0 ( x)dx (ii)

 J ( x)dx

(iii)

3

 x J ( x)dx . 3

1

10. Show that 4J n( x)  J n2 ( x)  2J n ( x)  J n2 ( x) . 11. Show that lim xJ 3/2 ( x)   x0

2



.



12. Using Rodrigue’s formula, prove that 13. Prove that Pn (1) 

1

P ( x)dx  0 .

1 n

n(n  1) . 2

1

1

1

1

1

1

1

14. Prove that: 𝑃𝑛 (− 2) = 𝑃0 (− 2) 𝑃2𝑛 (2) + 𝑃1 (− 2) 𝑃2𝑛−1 (2) + − − +𝑃2𝑛 (− 2) 𝑃0 (2) 15. Prove that (1 − 2𝑥𝑧 + 𝑧 2 )

−1⁄ 2

is a solution of the equation 𝑧

𝜕2 (𝑧𝑣) 𝜕𝑧 2

𝜕

+ 𝜕𝑥 {(1 −

𝜕𝑣

𝑥 2 ) 𝜕𝑥} = 0 16. If 𝑚 > 𝑛 − 1 and n is a positive integer, prove that 1

∫0 𝑥 𝑛 𝑃𝑛 (𝑥)𝑑𝑥 =

𝑚(𝑚−1)(𝑚−2)−−−−−(𝑚−𝑛−2) (𝑚+𝑛+1)(𝑚+𝑛−1)−−(𝑚−𝑛+3) 1

17. If 𝑚 > 𝑛, show that ∫−1 𝑥 𝑛 𝑃𝑛 (𝑥)𝑑𝑥 = 18. Deduce from Rodrigue’s formula:

2𝑛+1 (𝑛!)2

(2𝑛+1)! 1 ∫−1 𝑓(𝑥)𝑃𝑛 (𝑥)𝑑𝑥

=

(−1)𝑛 2𝑛 (𝑛!)

1

∫−1(𝑥 2 − 1)𝑛 𝑓 (𝑛) (𝑥)𝑑𝑥

19. Prove that: 𝐽0 2 + 2(𝐽1 2 + 𝐽2 2 + − − − − −) = 1 20. Prove that: 𝑥 = 2𝐽0 𝐽1 + 6𝐽1 𝐽2 + − − − + 2(2𝑛 + 1)𝐽𝑛 𝐽𝑛+1 + − −

21. Prove that: 𝐽𝑛 (𝑥) = 𝑑

(𝑥⁄2)

𝑛

√𝜋 𝑔𝑎𝑚𝑚𝑎

𝐽

22. Prove that: 𝑑𝑥 ( 𝐽−𝑛 ) = −

1 (𝑛+ ) 2

1

1

𝑛− ∫−1(1 − 𝑡 2 ) 2 𝑒 𝑖𝑥𝑡 𝑑𝑡

(𝑛 >

2𝑆𝑖𝑛 𝑛𝜋

𝑛

𝜋𝑥𝐽𝑛 2

8

4

23. Show that: 𝐽3 (𝑥) = (𝑥 2 − 1) 𝐽1 (𝑥) − 𝑥 𝐽0 (𝑥) 2

1

3

24. Show that: 𝐽5⁄ (𝑥) = √𝜋𝑥 [𝑥 2 (3 − 𝑥 2 )𝑆𝑖𝑛 𝑥 − 𝑥 𝐶𝑜𝑠 𝑥] 2

25. Show that 𝐶𝑜𝑠 (𝑥 cos 𝜃) = 𝐽0 − 2𝐽2 𝐶𝑜𝑠 2𝜃 + 2𝐽4 cos 4𝜃 − − − − and 𝑠𝑖𝑛 (𝑥 cos 𝜃) = 2[ 𝐽1 𝐶𝑜𝑠 𝜃 − 𝐽3 cos 3𝜃 − − −−]

−1 2

)

UNIT-2 Problem set 1 1. Eliminate the arbitrary constants to obtain a partial differential equation (i) (𝑥 − 𝑎)2 + 𝑦 2 + (𝑧 − 𝑏)2 = 16 (ii) 𝑧 = (𝑥 + 𝑎𝑦)2 + 𝑏𝑦 2. Eliminate the arbitrary function to obtain a partial differential equation 𝑥𝑦𝑧 = 𝑓(𝑥 + 𝑦 + 𝑧) 3. Eliminate the arbitrary function to obtain a partial differential equation 𝑧 = 𝑓(𝑥 + 𝑖𝑦) + 𝑔(𝑥 − 𝑖𝑦), 𝑖 2 = −1. 4. Find the differential equation of all planes which are at a constant distance ‘a’ from the origin. 5. Find the differential equation of all spheres whose centres lie on the z-axis. 6. Find the differential equation of all spheres of radius 3 units having their centres in the xy-plane. 7. Form partial differential equation by eliminating the arbitrary function from the given equation 𝑦 𝑧 = 𝑓( ) 𝑥 8. Form the partial differential equation by eliminating arbitrary constants: 1 𝑧 = 𝑎𝑥𝑒 𝑦 + 𝑎2 𝑒 2𝑦 + 𝑏 2 9. Form the partial differential equation by eliminating arbitrary constants: 𝑧 = 𝑥𝑦 + 𝑦√𝑥 2 − 𝑎2 + 𝑏 10. Form partial differential equation by eliminating the arbitrary function from the given equation 𝑓(𝑥𝑦 + 𝑧 2 , 𝑥 + 𝑦 + 𝑧) = 0 11. Form the partial differential equation by eliminating arbitrary constants: 2

𝑧 = 𝐴𝑒 −𝑝 𝑡 cos⁡(𝑝𝑥) 12. Form the partial differential equation by eliminating arbitrary constants: 𝑒

1 𝑥2 ) 𝑦

𝑧−(

=

𝑎𝑥 2 𝑏 + 𝑦2 𝑦

Problems set-2

1. Form the partial differential equation (by eliminating arbitrary functions) from a.

z  ( x  y)  ( x 2  y 2 )

b. z  f ( x  at )+g ( x  at ) c.

z  f ( x2  y2 , z  xy) .

2. Find the complete solution of the partial differential equation 3. Solve the partial differential equation

p2 1  q   qxy .

2 z 2 z 2 z  5  7  y sin x . x2 xy y 2

4. Find the complete solution of the partial differential equation

p 2  q 2  az .

2  2u 2  u c 5. Find the general solution of heat equation . t 2 x2

6. Find the complete solution of the partial differential equation

z  px  qy  tan pq .

7. A uniform rod of length L whose surface is thermally insulated is initially at temperature

T  T0 . Its one end is suddenly cooled to T  0 and subsequently maintained at this temperature; the other end remains thermally insulated. Find temperature distribution T ( x, t ) along the rod.

8. Find the complete solution of the partial differential equation

p2 x  qz  qxy .

u  2u  9. Find the general solution of heat equation . t x 2 10. A uniform string of line density  is stretched to tension  c 2 and excutes a small transverse vibration in a plane through the undisturbed line of string. The ends x  0 and

x  L of the string are fixed. The string at rest, with the point x  b drawn aside through a small distance  , is released at time t  0 . Find an expression for the displacement y ( x, t ) .

11. Solve the partial differential equation

2 z 2 z 2 z   6  y cos x . x2 xy y 2

L is held fixed at its ends and is subjected to an initial displacement u( x,0)  u0 sin( x L) . The string is released from its position with zero

12. A stretched string of finite length

initial velocity. Find the displacement of the string. 13. Find the complete solution of the partial differential equation

p 1  q   qz , where p 

z z ,q  . x y

Problems set-3

1.

 4z  4z   0 Ans:: z  f1 ( y  x)  f2 ( y  x)  f3 ( y  ix)  f4 ( y  ix) x 4 y 4

2.

 4z  4z  4z  4z  2   0 x 4 x 3 y xy 3 y 4

Ans:: z  f1 ( y  x)  f2 ( y  x)  xf3 ( y  x)  x 2 f4 ( y  x) 3.

1 2z 2z 2z  2  2  e 3x2 y Ans:: z  f1 ( y  x)  xf2 ( y  x)  e 3x 2 y 2 25 xy y x

4.

( x  y) 3 2z 2z 2z 3  2 2  x  y Ans:: z  f1 ( y  x)  xf2 ( y  2x)  36 xy x 2 y

5.

2z 2z 2z  2   sin(2x  3y) Ans:: z  f1 ( y  x)  xf2 ( y  x)  sin(2x  3y) xy y 2 x 2

2z 2z   cos mx cos ny  30(2x  y) x 2 y 2 1 cos mx cos ny  ( 2 x  y) 3 Ans:: z  f1 ( y  ix)  f2 ( y  ix)  2 2 m n 6.

7.

 3z  3z  3z  4  4  4 sin(2x  y) x 3 x 2 y xy 2

Ans: z  f1 ( y)  f2 ( y  2x)  xf3 ( y  2x)  x 2 cos( 2x  y) 8.

2z 2z x 1   sin x cos y Ans: z  f1 ( y)  f2 ( y  2 x)  cos( x  y)  sin(x  y) 2 xy 2 4 x

9.

D

10.

x6 y3 x9  3z  3z 2 3 3 z  f ( y  x )  f ( y  ω x )  f ( y  ω x )   Ans:   x y 1 2 3 120 10080 x 3 y 3

11.

2z 2z 2z   6  y cos x Ans: z  f1 ( y  2x)  f2 ( y  3x)  y cos x  sin x x 2 xy y 2

2



 DD' z  cos 2y(sin x  cos x) 1 1 Ans: z  f1 ( y)  f2 ( y  x)  cos( x  2 y)  sin(x  2 y)  cos( x  2 y)  sin(x  2 y) 2 6

2z 2z 2z   2  y  1e x Ans: z  f1 ( y  x)  f2 ( y  2x)  ( y  2)e x 2 2 xy x y 13. D  D'1D  2D'2 z  0 Ans: 12.

2z 2z 2z  3z x     2  y  1 e cos x  4  4 sin(2 x  y) cos mx cos ny  30( 2 x  y)x  ye 3x 2 y x 2 xy y 2 xy 2

x6 y3 x9 z  e f1 ( y  x)  e f2 ( y  x)  ( y  2 )e y cos x  sin x  f3 ( y  ω x)   cos( 2 x  y)f4 ( y  x) 120 10080 D  D'1D  2D'2 z  0 cos 2 y(sin x  cos x) x

2x

x

2

Problems set-4

1. Use the method of separation of variables to solve the equation that u( x,0)  6e  x .Ans: u( x , t )  6e 3 x2 t

2. Use the method of separation of variables to solve the equation



Ans: u(x, y)  Ae

1



1 p 2 x

 Be1



1 p 2 x

e

u u 2  u ,given x t

2u u u 2  0. 2 x x y

p2 y

2 2y 2  y 3. Transform the equation 2  c to its normal form using the transformation t x 2 u  x  ct , v  x  ct and hence solve it. Show that the solution may be put in the form y 1 y  f( x  ct )  f( x  ct ) . Assume the initial conditions y  f(x) and  0 at t=0. 2 t Ans: y  φ(x  ct )  ψ(x  ct )

4. Reduce the equation u xx  2u xy  u yy  0 to its normal form using the transformation

v  x, z  x  y and solve it. Ans:

2u  0 , u  xf1 (x  y)  f2 (x  y) v 2 Wave Equation

2y 2y 1. Show how the wave equation c can be solved by the method of separation  x 2 t 2 2

of variables. If the initial displacement and velocity of a string stretched between x=0 and x=l are given by y  f( x) and

y  g( x) . Determine the constants in the series t



nπct nπct  nπx where  b n sin  sin L L  L 1 L L 2 nπx 2 nπx a n   f( x) sin dx and b n  g(x) sin dx . Here L=l.  L0 L nπc 0 L 2. Find the deflection y( x, t ) of the vibrating string of length π and ends fixed, corresponding to zero initial velocity and initial deflection f(x)  k(sin x  sin 2x) , given Ans: y( x , t ) 

solution.

  a

n

cos

c 2 =1.Ans: y(x, t )  k(cos t sin x  cos 2t sin 2x)

3. A string is stretched and fastened to two points l apart. Motion is started by displacing x the string from the initial deflection y  A sin , from which it is released at time l t=0.Show that the displacement of any point at a distance x from one end at time t is given by y  A sin

x l

cos

ct l

.

4. If a string of length l is initially at rest in equilibrium position and each of its points is  y  3 x given the velocity  , find the displacement y( x , t ) .   b sin l  t  t 0 bl  x ct 3x 3ct  Ans: y( x , t )  . 9 sin sin  sin sin 12c  l l l l  5. A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0, the string is given a shape defined by f(x)  x(l  x) , μ is a constant and then released. Find the displacement y( x , t ) of any point x of the string at any time t>0. Ans: y( x , t ) 

8μL2 π3



1

 (2n  1) n 1

3

sin

( 2 n  1) πx ( 2 n  1) πct cos L L

Heat Equation

1. A rod of length L with insulated sides is initially at a uniform temperature u 0 .Its ends are suddenly cooled to 0 0 C and are kept at that temperature. Find the temperature

4u 0 function u(x,t) .Ans: u( x, t )  π

(2n  1)πx 1 sin e  L n 1 2 n  1 

 c 2 ( 2 n 1 )2 π 2 t L2

2. An insulated rod of length L has its ends A and B maintained at 0 0 C and 1000 C respectively, until steady state condition prevails. If B is suddenly reduced to 0 0 C and maintained at 0 0 C , find the temperature at a distance x from A at time t. Find also the temperature if the change consist of raising the temperature of A to20 0 C and reducing the temperature of B to 80 0 C .

60x 40  1 2mπx Ans: u( x, t )  20   sin e  L π m 1 m L

4 c 2 m 2 π 2 t L2

3. The initial temperature of an insulated infinite rod is given by u( x,0)  ( 1) n U between x=nc and x=(n+1)c where n  I Show that for t>0,

2p  1πx 4U  1 u(x, t )  sin e  π p 0 2p  1 c

 d 2 ( 2 p  1 )2 π 2 t c2

4. A bar with insulated sides is initially at temperature 0 0 C throughout. The end x=0 is u kept at 0 0 C and heat is suddenly applied at the end x=L, so that =A for x=L, where x A is a constant. Find the temperature function u(x,t).

2n  1πx 8AL   1n Ans: u(x, t )  Ax  2  sin e 2 π n 1 2n  1 L 5. Solve the equation

c 2 ( 2 n 1 )2 π 2 t 4 L2

u  2 u  with boundary conditions u(x,0)  3 sin nπx , u(0, t )  0 , t x 2 

u(L, t)  0 where 0
2

π 2t

sin nπx

n 1

2u 2u   0 subject to boundary conditions x 2 y 2 nπx u(0, y)  u(L, y)  u(x,0)  0 and u( x , a)  sin .Ans: L  nπy  sinh   L  sin nπx u( x, y)  nπa  L sinh   L 

6. Use separation of variables to solve

UNIT-3 1. Show that lim𝑧→0 (𝑧⁄ ) does not exist. 𝑧 2. Show that the function 𝑓 is continuous at the given point.

3. Show that the function 𝑓(𝑧) = Arg(𝑖𝑧) is discontinuous at 𝑧0 = 𝑖 . 4. The function 𝑓(𝑧) = |𝑧|2 is continuous at the origin. (a) Show that 𝑓 is differentiable at the origin. (b) Show that 𝑓 is not differentiable at any point 𝑧 ≠ 0. 5. Show that the function

is not differentiable at 𝑧 = 0 by letting ∆𝑧 → 0 first along the 𝑥 −axis and then along the line 𝑦 = 𝑥. 6. Show that the function 𝑓(𝑧) = 3𝑥 2 𝑦 2 − 6𝑖𝑥 2 𝑦 2 is differentiable along the coordinate axes. 7. If 𝑥 is real then |sin 𝑥| ≤ 1 . Is the result true if 𝑥 is replaced by 𝑧 = 𝑥 + 𝑖𝑦 ? 8. Prove that the function 𝑓(𝑧) = 𝑒 𝑧 , 𝑧 ∈ ℂ is periodic with period 2𝜋𝑖. 9. If 𝑓(𝑧) is a complex function with pure imaginary period 𝑖 , then what is the period of the function 𝑔(𝑧) = 𝑓(𝑖𝑧 − 2)? 10. Let 𝑓(𝑧) =

𝑒 𝑖𝑧 +𝑒 −𝑖𝑧 2

.

(a) Show that 𝑓 is periodic with real period 2𝜋. (b) Suppose that 𝑧 is real i.e. 𝑧 = 𝑥 + 0𝑖 . What is the well-known real function do you get?

2

2

11. Prove that for 𝑧 = 𝑥 + 𝑖𝑦 ∈ ℂ , the equality |𝑒 𝑧 | = 𝑒 |𝑧| holds if and only if 𝑦 = 0. 2

12. Prove that the set of complex numbers 𝑧 = 𝑥 + 𝑖𝑦 such that |𝑒 𝑧 | ≤ 1 is given by −𝑦 ≤ 𝑥 ≤ 𝑦. 13. Find the constants 𝑎, 𝑏, 𝑐 such that the function 𝑓(𝑧) = −𝑥 2 + 𝑥𝑦 + 𝑦 2 + 𝑖(𝑎𝑥 2 + 𝑏𝑥𝑦 + 𝑐𝑦 2 ) is analytic. Also, express 𝑓(𝑧) in terms of 𝑧. 𝑦

14. In a two dimensional fluid flow, the stream function is = tan−1 𝑥 , find the velocity potential 𝜙. 15. Prove that 𝑢 = 𝑥 2 − 𝑦 2 − 2𝑥𝑦 − 2𝑥 + 3𝑦 is harmonic. Find a function 𝑣 such that 𝑓(𝑧) = 𝑢 + 𝑖𝑣 is analytic. Also express 𝑓(𝑧) in terms of 𝑧. 16. An electrostatic field in the 𝑥𝑦 plane is given by the potential function 𝜙 = 3𝑥 2 𝑦 − 𝑦 3 , find the stream function and hence find complex potential. 17. If 𝑛 is real, show that 𝑟 𝑛 (𝑐𝑜𝑠𝑛𝜃 + 𝑖𝑠𝑖𝑛 𝑛𝜃) is analytic except possibly when 𝑟 = 0 and that its derivative is 𝑛𝑟 𝑛−1 [cos(𝑛 − 1)𝜃 + 𝑖𝑠𝑖𝑛( 𝑛 − 1)𝜃]. 18. Let 𝑓(𝑧) = 𝑢(𝑟, 𝜃) + 𝑖𝑣(𝑟, 𝜃) be an analytic function. If 𝑢(𝑟, 𝜃) = −𝑟 3 𝑠𝑖𝑛3𝜃 then construct the corresponding analytic function 𝑓(𝑧) in terms of 𝑧 . 19. If 𝜙 and 𝜓 are functions of 𝑥 and 𝑦 satisfying Laplace’s equation, show that 𝑠 + 𝑖𝑡 is analytic where 𝑠 =

𝜕𝜙 𝜕𝑦

𝜕𝜓

− 𝜕𝑥 and 𝑡 =

𝜕𝜙 𝜕𝑥

𝜕𝜓

+ 𝜕𝑦 .

20. Find an analytic function 𝑓(𝑧) such that 𝑅𝑒[𝑓 ′ (𝑍)] = 3𝑥 2 − 4𝑦 − 3𝑦 2 and 𝑓(1 + 𝑖) = 0. 21. Show that the function 𝑓(𝑧) defined by 𝑓(𝑧) =

𝑥 3 𝑦 5 (𝑥+𝑖𝑦) 𝑥 6 +𝑦 10

, 𝑧 ≠ 0; 𝑓(0) = 0, is not analytic at

the origin even though it satisfies Cauchy-Riemann equation at the origin. 22. If 𝑓 ′ (𝑧) = 𝑓(𝑧) for all 𝑧, then show that 𝑓(𝑧) = 𝑘𝑒 𝑧 , where 𝑘 is an arbitrary constant. 𝑢𝑥 23. Show that if 𝑓(𝑧) is differentiable at a point 𝑧, then |𝑓′(𝑧)|2 = | 𝑣 𝑥 24. Determine constant 𝑏 such that 𝑢 = 𝑒 𝑏𝑥 𝑐𝑜𝑠5𝑦 is harmonic.

𝑢𝑦 𝑣𝑦 |.

25. Show that (i) 𝑓(𝑧) = 𝑧 + 2𝑧̅ is not analytic anywhere in the complex plane (ii) 𝑓(𝑧) = 𝑥𝑦 + 𝑖𝑦 is everywhere continuous but is not analytic. 26. State and prove Cauchy- Integral theorem for multiple connected domain. 27. If f  z  is analytic inside a simple connected domain then show that integral of f  z  is path independent. 28. Using Cauchy integral formula Show that every bounded entire function must be constant. 29. Verify the Cauchy’s theorem by integrating at the points 1  i,  1  i,  1  i.

ei z

along the boundary of the triangle with the vertices

z 3 taken over the boundary of the (i) rectangle with vertices 1, 1, 1  i,  1  i; (ii) triangle with vertices 1, 2 , 1, 4 ,  3, 2 .

30. Verify the Cauchy’s theorem by integrating

31. Evaluate the followings using Cauchy integral formula:

sin  z 2  cos  z 2 dz, where c the circle is z  3 . 32.  c (z 1)(z 2)

e2 z dz, where c the circle is z  2 . 33.  c (z1)5 z2 1 dz, , where C is the circle z  1  1 . 34.  c z z 1 2   35.

e3i z c  z    dz, ,

36.

z3  z  1 c z 2  7 z  2 dz, ,

37.

where C is the circle z    3 .

e z  sin  z

where C is the ellipse 4 x2  9 y3  1 .

  z 1 z  3  z  1 dz, , c

2

where C is the circle z  2 .

38.



c

z

z

2

 6 z  25 2 3i

39. Evaluate



2

dz, , where C is the circle z  3  4i  4 .

( z 2  z )dz along the line joining the points (1,-1) and (2,3).

1i

2 i

40. Show that for every path between the limits



2

i (2  z )2 dz   . 3

41. Verify the Cauchy theorem for the integral of 𝑧 3 taken over the boundary of the rectangle with vertices -1, 1, 1+I and -1+i.

42. Evaluate the value of

ez dz 2  C ( z  1)

where C is the circle z  1  3 .

sin z 2  cosz 2 dz , where C is the circle z  3 . C ( z  1)( z  2) 44. Evaluate the following complex integration using Cauchy’s integral formula 43. Use Cauchy Integral formula to evaluate 

3z 2  z  1 C ( z 2 1)( z  3)dz where C is the circle z  3 . 45. Obtain the Taylor’s series expansion of f (z ) =

1 about z =4 and find its region of z  4z  3 2

convergence. 46. Expand the following function in a Laurent’s series f (z ) =

z 2 1 47. Expand the function f ( z )  in the regions ( z  2)( z  3) (i) 2  z  3

(ii)

z 3

1 for z  1  1 . z( z  1)( z  2)

UNIT-4

1. 2. 3. 4. 5. 6.

Singularities and Evaluation of integrals by Residue theorem cot z Discuss singularity of at z= a Ans: Double pole z  a2 1  Discuss singularity of at z  Ans: simple pole sin z  cos z 4 Find the poles and residue at each pole of the function cot z . Ans: z  n , n  I ,1 4 1  e2z Determine the poles of and residue at each pole. Ans: z  0 ,  4 3 z 1 1 Find the residue of z cos  at z  0 .Ans:  2 z z2 Determine poles and residue at each pole of the function f z  and hence z  12 z  2 5 5 4 evaluate  f zdz ,where C is the circle z  Ans: -1,2; , ; 2i 2 9 9 C

z 1 and hence z  12 z  2 1 1 2i evaluate  f zdz ,where C is the circle z  i  2 Ans: -1,2;  , ;  9 9 9 C

7. Determine poles and residue at each pole of the function f z 

z 2  2z dz ,where C is the circle z  10 Ans:0 8. Evaluate  z  12 z 2  4 C 9. Evaluate

12z  7

 z 1 2z  3 dz ,where C is the circle 2

z  i  3 Ans: 4i

C

z3 at pole and hence evaluate z  14 z  2z  3 5 27i C f zdz ,where C is the circle z  2 Ans:  8 sin z 11. Find the sum of residues of the function at its pole inside the circle z  2 . Ans: 0 z cos z 10. Find the residue of the function f z 

Evaluation of real integrals

ad  , a  0 .Ans: 2 a  sin  a2  1 2 cos 3 d  2. Evaluate  .Ans:  0 5  4 cos  12 2 d 2  2 2 , where a  b  0 3. Prove that  0 a  b cos  a b 1 1. Evaluate





0

2

d 2  2 2 , where a  b  0 a  b sin a b 2 2 sin  d 4i 5. Using complex variable technique ,evaluate  Ans: 3 0 5  4 cos 3 3 2 2 d 2 0  a  1 Ans: 6. Evaluate  0 1  2 a sin   a 2 1  a2  2 d d 7. Evaluate  , where a  b .Hence or otherwise evaluate  . Ans: 0 a  b cos  0 2  cos  4. Prove that



a2  b2



2

0

, 2

8. Apply calculus of residues to prove that

d

2

 a  b cos  

2

0

dx



 x



2a

a

 b 2 2 3

2

;a  0 3  a  4a  dx  2 10. Apply calculus of residues to prove that  4 4  ;a  0 0 x a 4a 3  x sin x  dx, a  0 Ans: e  a 11. Apply calculus of residues to evaluate  2 2 0 x a 2  cos mx  ma dx, m  0, a  0 Ans: e 12. Using contour integration, evaluate  2 2 0 x a 2a   e b e  a   cos x    13. Using contour integration, evaluate  2 2 2 2 dx, a  b  0 Ans: 0 x  a x  b  2a 2  b 2   b a   sin x 14. Using contour integration, evaluate  2 2 2 2 dx, a  b  0 Ans: 0 0 x  a x  b   sin mx  dx, m  0 Ans: 15. Evaluate  0 x 2  sin mx dx  16. Apply calculus of residues to prove that   2 1  e ma ; a  0 2 2 0 xx  a  2a 9. Apply calculus of residues to prove that

0

2

2

2 2





3