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 h3 5 y h 2 8 y h1 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 ayk 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 Vn1 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 Vn1 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 n2 ( x) 2J n ( x) J n2 ( x) . 11. Show that lim xJ 3/2 ( x) x0
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 xy 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 xy 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 xy 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 3x2 y Ans:: z f1 ( y x) xf2 ( y x) e 3x 2 y 2 25 xy y x
4.
( x y) 3 2z 2z 2z 3 2 2 x y Ans:: z f1 ( y x) xf2 ( y 2x) 36 xy x 2 y
5.
2z 2z 2z 2 sin(2x 3y) Ans:: z f1 ( y x) xf2 ( y x) sin(2x 3y) xy 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 xy 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 xy 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 xy 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 1e x Ans: z f1 ( y x) f2 ( y 2x) ( y 2)e x 2 2 xy x y 13. D D'1D 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 xy y 2 xy 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'1D 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 x2 t
2. Use the method of separation of variables to solve the equation
Ans: u(x, y) Ae
1
1 p 2 x
Be1
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 3x 3ct Ans: y( x , t ) . 9 sin sin sin sin 12c 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 1πx 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 1πx 8AL 1n 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<x
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)
e2 z dz, where c the circle is z 2 . 33. c (z1)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).
1i
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 cosz 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 a2 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 12 z 2 5 5 4 evaluate f zdz ,where C is the circle z Ans: -1,2; , ; 2i 2 9 9 C
z 1 and hence z 12 z 2 1 1 2i evaluate f zdz ,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 12 z 2 4 C 9. Evaluate
12z 7
z 1 2z 3 dz ,where C is the circle 2
z i 3 Ans: 4i
C
z3 at pole and hence evaluate z 14 z 2z 3 5 27i C f zdz ,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 4i 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
2a
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 2a 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 xx a 2a 9. Apply calculus of residues to prove that
0
2
2
2 2
3