Class notes EMA51001 (Maths)

DEPARTMENT OF MATHEMATICS


COURSE NAME: ANALYTICAL MATHEMATICS
COURSE CODE: EMA51002




QUESTION BANK
WITH SOLUTIONS

, DEPARTMENT OF MATHEMATICS
Subject: EMA51002-ANALYTICAL MATHEMATICS
End Semester Examination-Question Bank

PART-A
1. Find grad φ, if φ = 𝑥2+y2+z2
at (1, 1, -1)
2. Find the directional derivative of φ = 𝑥𝑦 + 𝑦𝑧 + 𝑦𝑧 at the

point (1,2,3) in the direction 3𝚤⃗ + 4 𝚥 + 5 𝑘
3. Define irrotational vector and solenoidal vector
4. State Stoke’s Theorem
5. State Green’s theorem
6. Prove that the vector 𝐹 =yz𝚤⃗ +𝑥𝑧𝚥 +xy𝑘 is solenoidal
7. State Cauchy Riemann equations
8. Check whether 𝑤 = 𝑒 𝑧 is analytic everywhere
1− z
9. Find the invariant point of the bilinear transformation w =
1+ z
10. Is f (z) = z 2 analytic? Justify
11. State the necessary conditions for 𝑓(𝑧) to be analytic
12. Prove that f(z)=sinz is analytic
13. Define Cauchy’s residue theorem
14. State Cauchy’s integral theorem
15. Expand f (z) = e as Taylors series at z=0
z

𝑧+2
16. Find the poles of 𝑓(𝑧) = and identify its degree
(𝑧−1)(𝑧−3)
17. Define Taylor series
𝑧+2
18. Define pole and find the poles of (𝑧) =
(𝑧−1)(𝑧−3)
19. Find the Laplace transform for (i) 𝑒7𝑡 (ii) 2𝑡
20. Find the Laplace transform of [𝑐𝑜𝑠ℎ2𝑡 − 𝑐𝑜𝑠2𝑡]
21. Find the Laplace transform for (i) 𝑒7𝑡 (ii) 𝑡4
22. State the first shifting theorem in Laplace transformation
23. State initial and final value theorem
1
24. Find the inverse Laplace transform for (𝑠+3)4
25. State Dirichlet’s condition
26. Find the value of a0 for the function f(x) = x2 defined in the interval (0,2π)
27. Find the constant 𝑎0 of the Fourier series for the function 𝑓(𝑥) = 𝑥 + 1 in
0 ≤ 𝑥 ≤ 2𝜋
28. State Dirichlet’s condition

,29. Find the value of a0 in the Fourier series expansion of f(x) = ex in (0,2π)
30. What do you mean by Harmonic Analysis?

PART-B
31. Find the constants a, b and c such that
𝐹 =(𝑎𝑥𝑦 + 𝑏𝑧3)𝚤⃗ +(3𝑥2 − 𝑐𝑧) 𝚥 +(3𝑥𝑧2 − 𝑦 ) 𝑘 is irrotational
32. Find the angle between the surfaces𝑥2 − 𝑦2 − 𝑧2 = 11 and (𝑥𝑦 + 𝑦𝑧 − 𝑧𝑥) = 18 at
(6,4,3).
33. Using Green’s theorem in the plane evaluate, ∫𝐶 (3𝑥2 − 8𝑦2) 𝑑𝑥 + (4𝑦 − 6𝑥𝑦)𝑑𝑦,
where 𝐶 is the boundary of the region enclosed by y=√𝑥 and y=𝑥2
34. Find the angle between the surfaces 𝑧 = 𝑥2 + 𝑦2 − 3 and 𝑥2 + 𝑦2 + 𝑧2 = 9 at (2,-1,2)
35. Evaluate ∫𝐶 𝑥𝑦𝑑𝑥 + 𝑥𝑦2𝑑𝑦, where C is the square in xy plane with vertices (1,0), (-1,0),
(0,1), and (0,-1), using Stoke’s theorem
36. Apply Green’s theorem in the plane evaluate ∫𝐶 (3𝑥2 − 8𝑦2) 𝑑𝑥 + (4𝑦 − 6𝑥𝑦)𝑑𝑦,
where C is the boundary of the region defined by x=0,y=0 and x+y=1
37. Construct an analytic function whose imaginary part is given by v = e x sin y
38. Find the analytic function whose real part is 𝑢 = 2𝑥𝑦 + 3𝑦
39. Construct the analytic functions whose real part is given by
u = x − 3xy + 3x − 3y +1
3 2 2 2

40. What is the region of the w-plane into the rectangular region in z-plane bounded by the
lines x = 0, x = 2, y = 0, y = 1 in the region mapped under the transformation
w = z + (1 + 2i)
41. Find the bilinear transformation that maps the points z=0,-1,i into the points w=i,0,∞
42. Find the image of the circle z = a under the transformation w = z +
𝑧
43. Evaluate∫ 𝑒
𝑐 𝑧−1
𝑑𝑧 ; 𝑐 is the circle |𝑧| = 2
44. Obtain the Taylor series to represent the function 1
in the region |𝑧| < 2
(𝑧+2)(𝑧+3)
45. Obtain the Taylor series to represent the function 𝑓(𝑧) = log (1 + 𝑧) about z = 0 if
|𝑧| < 1
46. Expand f(z)=sin z using Taylors series at z=π/4
2z 2 − 4z + 3
47. Evaluate the following using Cauchy integral formula 
c
z−2
𝑑𝑧 where 𝐶 is

|𝑧| = 3
z2 + 2
48. Evaluate the following using Cauchy integral formula  c z−2
𝑑𝑧 where 𝐶 is the circle

|𝑧| = 3
49. Find the Laplace transform for 𝑒𝑡 sin 4𝑡 cos 2𝑡
50. Evaluate 𝐿[𝑒−𝑡𝑐𝑜𝑠4𝑡]
51. Verify initial value theorem for 𝑓(𝑡) = 2 + 3 𝑐𝑜𝑠𝑡
52. Find the inverse Laplace transform for 𝑆+2
(𝑆+4)2

, 𝑠
53. Using partial fraction method, find 𝐿−1 [ ]
(𝑠+2)(𝑠+3)
54. Find the inverse Laplace transforms for 1
(𝑠+1)(𝑠+2)
55. Find the Fourier series of the function 𝑓(𝑥) = 𝑥 in the interval (0,2π)
56. Find the Fourier series of the function 𝑓(𝑥) = 𝑥 in the interval (-π, π)
57. Find the Fourier series for 𝑓(𝑥) = |𝑥| in (−𝜋, 𝜋)
58. Find the half range Fourier cosine series of 𝑓(𝑥) = 𝑒𝑥 defined in the interval (0,π)
59. Find the half range cosine series for 𝑓(𝑥) = 𝑥( 𝜋 − 𝑥) in (0, 𝜋)
60. Find the half range Fourier cosine series of 𝑓(𝑥) = 𝑒𝑥 defined in the interval (0, π)
PART-C
61. Verify green’s theorem for 𝑐∫ (𝑥2
𝑑𝑥 + 𝑥𝑦 𝑑𝑦) where C is a rectangle bounded by the
line 𝑥 = 0, 𝑦 = 0, 𝑥 = 𝑎, 𝑦 = 𝑏
62. Verify Gauss divergence theorem for 𝐹 = 4𝑥𝑧𝚤⃗ -𝑦2𝚥 + 𝑦 𝑧 𝑘 over the cube bounded by
x=0, x=1, y=0, y=1, z=0, z=1
63. Verify Gauss divergence theorem for 𝐹̅= 𝑥2𝚤⃗ + 𝑦2𝚥 + 𝑧 2 𝑘 taken over the cube
bounded by the line 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 = 1, 𝑦 = 1, 𝑧 = 1
64. Verify Gauss divergence theorem for the function 𝐹 = (𝑥2 − 𝑦𝑧)𝚤⃗ +(𝑦2 − 𝑧𝑥)𝚥 +
(𝑧2 − 𝑥𝑦 ) 𝑘 taken over the parallelopiped 0 ≤ 𝑥 ≤ a, 0 ≤ 𝑦 ≤b, 0 ≤ 𝑧 ≤c
65. Verify Stoke’s theorem for 𝐹 = (𝑥2 + 𝑦2)𝚤⃗ − 2𝑥𝑦𝚥 taken around the rectangle
bounded by the lines 𝑥 = ±𝑎, 𝑦 = 0,𝑦 = 𝑏
66. Verify Stoke’s theorem for 𝐹 = (𝑥2 − 𝑦2)𝚤⃗ + 2𝑥𝑦𝚥 for rectangular region bounded by
𝑥 = 0, 𝑥 = 𝑎, 𝑦 = 0, 𝑦 = 𝑏 on XY Plane
67. Find the analytic function whose real part is sin 2𝑥
cosh 2𝑦−cos 2𝑥
y
68. Construct an analytic function f (z) , whose real part is u =
x + y2
2


69. Find the bilinear transformation that maps z1 = 0, z 2 =1, z 3 = onto
1 =− 1, 2 =− i, 3 =1
70. Find the bilinear transformation that maps Z=-1,0,1 onto w =-1,-1,1
1
71. Find the image of z − 2i under the transformation w =
z
1
72. Obtain the Laurents’s series for 𝑓(𝑧) = for (i) 1 < |𝑧| < 3
(𝑧+1)(𝑧+3)

ii) |𝑧| < 1
73. Find the Laurent’s series expansion of the function 1
valid for the regions
(𝑧+1)(𝑧+3)
|𝑧| > 3 and 1 < |𝑧| < 3
1
74. Obtain the Laurent’s series for 𝑓(𝑧) = for (i) 0 < |𝑧| < 1 (ii) 0 < |𝑧 − 1| < 1
𝑧(𝑧−1)
𝟐𝝅 𝒅𝜽
75. Evaluate using contour integration, ∫ 𝟎 𝟓−𝟑 𝐜𝐨𝐬 𝜽
2𝜋 𝑑𝜃
76. Evaluate 0∫
5+4𝑠𝑖𝑛𝜃
77. Find the Laplace transform of the periodic function defined by