Question Bank – Engineering Mathematics, Volume III

Question Bank

Mulitple Choice Questions

Chapter 1 Special Functions

A. Beta and Gamma Functions

  1. If Γ(x) = 1, then x =
    1. 1
    2. 0
    3. −1/2
    4. 1/2

    Ans: (a)

  2. Γ(1/2) =
    1. π/2

    Ans: (c)

  3. If p = 1/4, then Γ(p) Γ(1 – Γp) =

    Ans: (d)

    1. 2/3
    2. 1/3
    3. 3/2
    4. 3

    Ans: (b)

    1. π/2

    Ans: (a)

  4. If Γ(2p) = aΓ(p) Γ(p + 1/2), then a =
    1. 22p–1
    2. 22p+1
    3. 22p
    4. 2p–1

    Ans: (a)

  5. B(m + 1, n)/B(m, n) =
    1. n/(m + n)
    2. (m + n)/m
    3. m/(m + n)
    4. (m + n)/n

    Ans: (c)

  6. Γ(p) Γ(1 – p) = πp =
    1. 1/4
    2. 2
    3. 1/2

    Ans: (d)

    1. Г(n)/an
    2. Г(n)/a
    3. Г(n + 1)/an
    4. Г(n)/an−1

    Ans: (a)

    1. 2
    2. 1
    3. 0

    Ans: (a)

    1. Г(p)
    2. Г(1/p)
    3. Г(p2)
    4. Г(1 − p)

    Ans: (b)

    1. 3π/2
    2. π/2
    3. π
    4. 0

    Ans: (c)

    1. π
    2. π/2

    Ans: (d)

    1. 2π
    2. π/2

    Ans: (b)

    1. π/2
    2. π

    Ans: (a)

B. Legendre Functions

    1. 2/(2n – 1)
    2. 2/(2n + 1))
    3. 1/(2n + 1)
    4. 1/(2n – 1)

    Ans: (b)

  1. The generating function for Pn(x) is
    1. (1 – 2xtt2)–1/2
    2. (1 – 2xt + t2)1/2
    3. (1 – 2xt + t2)–1/2
    4. (1 + 2xt + t2)–1/2

    Ans: (c)

  2. 1/(1 + i) = a + bi = ⇒ (a, b) =
    1. (1, –1)

    Ans: (a)

  3. P0 (x)
    1. 1
    2. 3/2
    3. 0
    4. 2/3

    Ans: (a)

    1. 0
    2. 1
    3. –1
    4. 1/2

    Ans: (c)

  4. By Rodrigue’s formula, Pn(x) = CDn{(x2–1)n} where the constant C =
    1. 1/[2n(2n)!]
    2. 1/n!
    3. 1/2n
    4. 1/(2n n!)

    Ans: (d)

  5. (1 + x) when expressed in terms of Legendre polynomials
    1. P0(x) – P1(x)
    2. P0(x) + P1(x)
    3. 2P0(x) – P1(x)
    4. P0(x) + 2P1(x)

    Ans: (b)

  6. xPn–1(x) + n Pn–1(x) =
    1. Pn–1(x)
    2. Pn(x)
    3. P′n(x)
    4. Pn+1(x)

    Ans: (c)

  7. The degree of the polynomial P4(x) is
    1. 3
    2. 2
    3. 4
    4. 1

    Ans: (c)

  8. Pn(–x) = anPn(x) where a =
    1. 2
    2. 0
    3. 1
    4. –1

    Ans: (d)

  9. (3x2 – 1) when expressed in Legendre polynomials
    1. P2 + 2P1
    2. 3P2P1
    3. 2P2
    4. 3P2P1

    Ans: (c)

    1. 1
    2. 2
    3. 0
    4. 3

    Ans: (b)

    1. 2/(2n + 1)
    2. 0
    3. 2/(2m + n)
    4. 2/(m + n)

    Ans: (b)

  10. P0 (x)
    1. 1
    2. 3/2
    3. 0
    4. 2/3

    Ans: (a)

    1. 0
    2. 1
    3. –1
    4. 1/2

    Ans: (c)

  11. By Rodrigue’sformula Pn(x) = CDn{(x2–1)n} where the constant C =
    1. 1/[2n(2n)!]
    2. 1/n!
    3. 1/2n
    4. 1/(2n n!)

    Ans: (d)

    1. 2/(2n – 1)
    2. 2/(2n + 1))
    3. 1/(2n + 1)
    4. 1/(2n – 1)

    Ans: (b)

  12. The generating function of Legendre function Pn(x) is
    1. (1 – 2xtt2)–1/2
    2. (1 – 2xt + t2)1/2
    3. (1 – 2xt + t2)–1/2
    4. (1 + 2xt + t2)–1/2

    Ans: (c)

C. Bessel Functions

    1. xn−1Jn−1
    2. xnJn−1(x)
    3. xn+1Jn(x)
    4. xn−1Jn(x)

    Ans: (b)

    1. x−nJm
    2. x−n+1Jn(x)
    3. x−nJn+1(x)
    4. x−nJn+1(x)

    Ans: (c)

  1. J1/2(x) =

    Ans: (a)

  2. Jn–1(x) – Jn+1(x) =
    1. Jn(x)
    2. 2Jn(x)
    3. 2Jn(x)
    4. Jn(x)

    Ans: (c)

    1. xnJn+1(x)
    2. xnJn−1(x)
    3. xnJn−1(x)
    4. Jn−1(x)

    Ans: (d)

  3. Jn(–x) =
    1. (–1)nJn(x)
    2. Jn(x)
    3. (–1)n–1Jn(x)
    4. (–1)nJn–1(x)

    Ans: (a)

    Ans: (a)

    1. cosec x
    2. cot x
    3. cos x
    4. sec x

    Ans: (b)

  4. J1(0) =
    1. 0
    2. 1
    3. –1
    4. 2

    Ans: (a)

  5. J0(0) =
    1. 1
    2. 0
    3. –1
    4. x

    Ans: (b)

  6. J0(x) is a solution of
    1. (xy′)′+ xy = 0
    2. (y′/x)′ + y = 0
    3. (x2y′) + xy = 0
    4. x(y′/x)′ + y = 0

    Ans: (a)

  7. J1/2(x) = p(x) sin x where p(x) =

    Ans: (c)

  8. [xnJn(x)]′ =
    1. xnJn+1(x)
    2. xnJn−1(x)
    3. xnJn−1(x)
    4. Jn−1(x)

    Ans: (c)

    1. 1/(πx)
    2. πx/2
    3. 2πx
    4. 2/(πx)

    Ans: (d)

    1. xJ1(x) − J0(x)
    2. xJ1(x)
    3. J1(x)
    4. x2Jn(x)

    Ans: (b)

    Ans: (a)

    1. J1(x)
    2. J1(x)
    3. (–1)nJn(x)
    4. xJ1(x)

    Ans: (b)

  9. [xJ1(x)]′ =
    1. 2J2(x)
    2. (−1)J1(x)
    3. xJ0(x)

    Ans: (c)

  10. Jn(–x) =
    1. Jn(x)
    2. (–1)nJn(x)
    3. 2Jn(x)
    4. J–n(x)

    Ans: (b)

    1. Jn(x)
    2. J–n(x)
    3. (–1)nJn(x)
    4. Jn(–x)

    Ans: (a)

Chapter 2 Functions Of A Complex Variable

  1. f (z) = (z + i)/(zi) is analytic
    1. for all z
    2. for all zi
    3. for all z ≠ – i
    4. for no z

    Ans: (b)

  2. f (z) = is analytic
    1. at z = 0 only
    2. everywhere
    3. nowhere
    4. if z ≠ 0

    Ans: (c)

  3. f (z) = |z|2 is analytic
    1. at z = 0 only
    2. everywhere
    3. nowhere
    4. if z ≠ 0

    Ans: (a)

  4. The harmonic conjugate of u(x, y) = x2y2 is v =
    1. x3y3
    2. x2 + y2
    3. 2xy
    4. x2y2

    Ans: (c)

  5. The analytic function whose real part is u(x, y) = x2y2 is f (z) =
    1. |z|2
    2. z3
    3. z2
    4. 2

    Ans: (c)

  6. The analytic function whose imaginary part is is f (z) =

    Ans: (b)

  7. Cauchy–Riemann equation in polar coordinates

    Ans: (c)

  8. Singularity is a point where f (z) is not
    1. defined
    2. having the limit
    3. continuous
    4. differentiable

    Ans: (d)

  9. The singularities of cosec z are at z =
    1. (n + 1/2)π
    2. (2n – 1)π
    3. 2

    Ans: (d)

  10. f (z) = ez has a singularity at
    1. the origin
    2. z = πi
    3. z = nπi
    4. no point

    Ans: (d)

  11. If f (z) = z/|z| (z ≠ 0) and f (0) = 1, then f is
    1. continuous of z = 0 only
    2. continuous for all z
    3. discontinuous for all z
    4. discontinuous at z = 0 only

    Ans: (d)

  12. If u(x, y) = 2xx2 + my2 is harmonic in D, then m =
    1. 0
    2. 1
    3. 2
    4. 3

    Ans: (b)

  13. If f (z) = z, then f′(z) exists
    1. for all z
    2. nowhere
    3. at z = 0 only
    4. at z = 1 only

    Ans: (c)

  14. A function u(x, y) having continuous second partial derivatives and satisfying ∇2u = 0 is called a
    1. harmonious function
    2. harmonic function
    3. holomorphic function
    4. regular function

    Ans: (b)

  15. The analytic function whose imaginary part is v(x, y) = 2xy is
    1. |z|2
    2. z
    3. l/z2
    4. z2

    Ans: (d)

  16. The harmonic conjugate of v(x, y) = 2xy is
    1. x2y2
    2. xy
    3. x2 + y2
    4. x3y3

    Ans: (a)

  17. 1/(1 + i) = a + bi ⇒ (a, b) =
    1. (1/2, –1/2)
    2. (–1, 1)
    3. (1,–1)
    4. (–1/2, 1/2)

    Ans: (a)

  18. (1 + i)/(1 – i) = a + bi ⇒ (a, b) =
    1. (1,–1)
    2. (–1, 1)
    3. (0, 1)
    4. (1, 0)

    Ans: (c)

  19. Re (ez) =
    1. ex
    2. e–x
    3. ex cos y
    4. e–x cos y

    Ans: (c)

  20. The complex conjugate of (1 + i)2 is
    1. i
    2. 2/i
    3. 2i
    4. 2 + i

    Ans: (b)

  21. A function u(x, y) having continuous second partial derivatives and satisfying ∇2u = 0 is called a
    1. harmonious function
    2. harmonic function
    3. holomorphic function
    4. regular function

    Ans: (b)

  22. The analytic function whose imaginary part is v(x, y) = 2xy is
    1. |z|2
    2. z
    3. l/z2
    4. z2

    Ans: (b)

  23. The harmonic conjugate of v(x, y) = 2xy is
    1. x2y2
    2. xy
    3. x2 + y2
    4. x3y3

    Ans: (a)

  24. Re (ez) =
    1. ex
    2. ex
    3. ex cos y
    4. e–x cos y

    Ans: (c)

  25. If f (z) = z/|z| (z ≠ 0) and f (0) = 1, then f is
    1. continuous of z = 0 only
    2. continuous for all z
    3. discontinuous for all z
    4. discontinuous at z = 0 only

    Ans: (d)

  26. If u(x, y) = 2xx2 + my2 is harmonic in D, then m =
    1. 0
    2. 1
    3. 2
    4. 3

    Ans: (b)

  27. If f (z) = z, then f′(z) exists
    1. for all z
    2. nowhere
    3. at z = 0 only
    4. at z = 1 only

    Ans: (c)

  28. The complex conjugate of (1 + i)2 is
    1. i
    2. 2/i
    3. 2i
    4. 2 + i

    Ans: (b)

  29. If f (z) = z(2 – z), then f (1 + i) =
    1. 0
    2. i
    3. i
    4. 2

    Ans: (b)

  30. If f (z) = |z| then f (3 – 4i) =
    1. 0
    2. 5
    3. –5
    4. 12

    Ans: (b)

  31. e2nπi =
    1. 0
    2. –1
    3. 1
    4. i

    Ans: (b)

  32. If f (z) = u + iv is analytic then
    1. ux = Vy, uy = Vx
    2. ux = Vy, uy = –Vx
    3. ux = –Vy, uy = Vx
    4. ux = Vx, uy = –Vy

    Ans: (c)

Chapter 4 Complex Integration

  1. A line integral of any complex function depends
    1. only on the initial point of the path
    2. only on the terminal point of the path
    3. only on the end points of the path
    4. on the end points as well as the choice of the path

    Ans: (d)

  2. The line integral of a complex function is independent of path if
    1. the function is analytic in a domain containing the path
    2. the domain is simply-connected
    3. the function is analytic in a simply-connected domain containing the path
    4. the function is continuous in a domain containing the path

    Ans: (c)

  3. The integral of a complex function f (z) vanishes over a path C if f (z) is
    1. analytic and C is any curve in domain D
    2. any complex function and C is any curve in domain D
    3. non-analytic but C is any closed path
    4. analytic and C is a closed path in a simply connected domain D

    Ans: (d)

  4. A bounded domain D is one which lies
    1. between two parallel lines
    2. in some circle about the origin
    3. outside the unit circle |z| = 1
    4. between a pair of intersecting lines

    Ans: (b)

  5. Among the following results, the one that does not follow from Cauchy’s Integral Theorem is

    Ans: (d)

  6. where C lies in . Cauchy’s Theorem is not applicable because
    1. f (z) ∉ H(D)
    2. D is not simply-connected though f (z)∊ H(D)
    3. f (z) ∊ H(d) but f (z) is not single-valued
    4. f (z) ∉ H (nbd of zero)

    Ans: (b)

  7. The value of over a simple closed curve C enclosing πi is
    1. –2 sinh π
    2. π sinh π
    3. 2 π sinh π
    4. 2 πi sinh π

    Ans: (c)

  8. If C is the circle |z| = 4, then the value of the integral of over C (counterclockwise) is
    1. 0
    2. –2πi
    3. 2πi
    4. 1

    Ans: (b)

  9. If over positively-oriented simple closed curve C about ‘a’, then m =
    1. –1
    2. 0
    3. ≠ –1
    4. 1

    Ans: (a)

  10. If C is the st. line segment from 0 to 1 + i, then an upper bound (by ML-inequality) for the absolute value of the integral of f (z) = z2 over C is
    1. 2
    2. 1

    Ans: (c)

  11. If C is a simple closed curve enclosing the origin, then
    1. πia
    2. 2πia
    3. 2πa

    Ans: (b)

  12. If C is a simple closed curve enclosing the origin, then
    1. 2π
    2. π
    3. –2πi
    4. 2πi

    Ans: (c)

  13. where C is a closed path, is
    1. 0
    2. πi
    3. 2πi
    4. 2π

    Ans: (c)

  14. If C is any simple closed path is true for f (z) =
    1. sec z
    2. ez

    Ans: (c)

  15. is true if C is
    1. |z| = 2
    2. |z| = π
    3. |z| = 1

    Ans: (d)

  16. where C : |z|=1.5, is

    Ans: (a)

    Ans: (b)

    1. 2i sinh π
    2. 2 sinh π
    3. 2i sin π

    Ans: (a)

    1. 1
    2. 2πi
    3. i
    4. 0

    Ans: (d)

    1. 2πi
    2. πi
    3. 4πi

    Ans: (c)

    Hint:

  17. is true if C is the circle
    1. |z| = 1
    2. |z| = 2
    3. |z| = 4
    4. none of these

    Ans: (a) or (b)

  18. If C is the straight line from z = 0 to z = i, then
    1. i/2
    2. 1/2
    3. i
    4. i/2

    Ans: (a)

  19. By ML-inequality, if C is the straight line from 0 to 1 + i we have
    1. 2
    2. 4

    Ans: (a)

    1. 2 sinh πi + 2πi
    2. zi sinh πi + π
    3. 2 sinh π + πi
    4. 2i(sinh π + π)

    Ans: (d)

    1. 2
    2. –2
    3. √2
    4. π

    Ans: (b)

  20. If C is the unit circle C : |zi| = 1 touching the z-axis at the origin, then
    1. 0
    2. 2πi
    3. πi
    4. πi/2

    Ans: (a)

  21. Among the following, which one is an entire function?
    1. ez/z
    2. z ez
    3. sec z
    4. z cosec z

    Ans: (b)

  22. The singularities of (z2 +1)/(z2 – 1) are at z =
    1. ±1
    2. ±i
    3. 1, i
    4. –1, –i

    Ans: (a)

  23. If where C is the circle |z| = 2, then f (3) =
    1. πi
    2. 2πi
    3. –1
    4. 0

    Ans: (d)

  24. If where C is the circle |z| = 2, then f (1) =
    1. 2πi
    2. πi
    3. 4πi
    4. 0

    Ans: (c)

  25. If C is the circle |z| = π traced in counterclockwise direction
    1. 2σ
    2. 2σi
    3. σi
    4. 4σi

    Ans: (b)

  26. If C is the unit circle |z| = 1, then
    1. 0
    2. 1
    3. 2πi
    4. 4πi

    Ans: (a)

  27. If C is the unit circle |z| = 1, then
    1. 0
    2. 1
    3. 2πi
    4. 4πi

    Ans: (c)

  28. If C is the unit circle |z| = 1 described in the counter-clockwise direction, then
    1. πi
    2. 2πi
    3. π/4
    4. π/8

    Ans: (d)

  29. If C is the circle |z| = 2 described in the anticlockwise direction, then
    1. 1
    2. –1
    3. 0
    4. 2

    Ans: (b)

    1. π
    2. i π/2
    3. i π
    4. 0

    Ans: (c)

  30. The order of the pole of (ez – 1)/z4 is
    1. 3
    2. 1
    3. 2
    4. 4

    Ans: (a)

  31. The residue of f (z) = z2/(z − 1)2(z + 2) at the simple pole z = –2 is
    1. 0
    2. 1
    3. 2/9
    4. 4/9

    Ans: (d)

    1. 2πi
    2. 0
    3. 2π
    4. 4πi

    Ans: (b)

  32. If C is the unit circle |z| = 1, then
    1. 2π
    2. π
    3. π
    4. –2π

    Ans: (a)

    1. 2πi
    2. πi
    3. πi

    Ans: (b)

    1. π
    2. –1
    3. 0
    4. 1

    Ans: (c)

  33. Along AB: y = 0, 0 ≤ x ≤ 1,
    1. 1/3
    2. 1/2
    3. 2/3
    4. 1

    Ans: (a)

  34. The set {zC/1 π |z| π 2} is a
    1. domain
    2. closed region
    3. simply-connected
    4. none of these

    Ans: (a)

    1. sin πi
    2. i sinh π
    3. i sin π
    4. sinh π

    Ans: (a) or (b)

    Ans: (a)

    1. π
    2. i π/2
    3. i π
    4. 0

    Ans: (b)

  35. The order of the pole of (ez – 1)/z4 is
    1. 3
    2. 1
    3. 2
    4. 4

    Ans: (a)

  36. The residue of f (z) = z2/(z − 1)2(z + 2) at the simple pole z = –2 is
    1. 0
    2. 1
    3. 2/9
    4. 4/9

    Ans: (d)

    1. 2πi
    2. 0
    3. 2π
    4. –2π

    Ans: (b)

  37. If C is the unit circle |z| = 1, then
    1. 2π
    2. π
    3. π
    4. –2π

    Ans: (b)

  38. If C is the circle |z| = π traced in counter clockwise direction
    1. 2π
    2. 2πi
    3. πi
    4. 4πi

    Ans: (b)

  39. If C is the unit circle |z| = 1, then
    1. 0
    2. 1
    3. 2πi
    4. 4πi

    Ans: (a)

  40. If C is the unit circle |z| = 1, then
    1. 0
    2. 1
    3. 2πi
    4. 4πi

    Ans: (c)

  41. If C is the unit circle |z| = 1 described in the counter-clockwise direction, then
    1. πi
    2. 2πi
    3. π/4
    4. π/8

    Ans: (d)

  42. If C is the circle |z| = 2 described in the anticlockwise direction, then where a =
    1. 1
    2. –1
    3. 0
    4. 2

    Ans: (b)

  43. If C is the unit circle C : |zi| = 1 touching the x-axis at the origin, then
    1. 0
    2. 2πi
    3. πi
    4. πi/2

    Ans: (a)

  44. Among the following, which one is an entire function?
    1. ez/z
    2. zez
    3. sec z
    4. zcosec z

    Ans: (b)

  45. The singularities of (z2 +1)/(z2 –1) are at z =
    1. ±1
    2. ±i
    3. 1, i
    4. –1, –i

    Ans: (a)

  46. If where C is the circle |z| = 2, then f (3) =
    1. πi
    2. 2πi
    3. –1
    4. 0

    Ans: (d)

  47. If where C is the circle |z| = 2, then f (1) =
    1. 2πi
    2. πi
    3. 4πi
    4. 0

    Ans: (c)

  48. is true if C is the circle
    1. |z| = 1
    2. |z| = 2
    3. |z| = 4
    4. none of these

    Ans: (b)

  49. If C is the straight line from z = 0 to z = i, then
    1. i/2
    2. 1/2
    3. i
    4. i/2

    Ans: (a)

  50. By ML-inequality, if C is the straight line from 0 to 1 + i we have
    1. 2√2
    2. 2
    3. √2
    4. 4

    Ans: (a)

    1. 2 sinh πi
    2. zi sinh πi
    3. 2 sinh π
    4. 2i sinh π

    Ans: (d)

    1. 2
    2. –2
    3. √2
    4. π

    Ans: (b)

Chapter 5 Complex Power Series

  1. The power series represents the function
    1. e(z–i)
    2. log(zi)

    Ans: (d)

  2. The series represents the function
    1. sin z
    2. cos z
    3. tan z
    4. sinh z

    Ans: (b)

  3. Taylor’s series expansion for f (z)=(z–1)/(z+1) about z = 0 is
    1. 1 + 2z + 2z2 + 2z3 + …
    2. –1 + zz2 + z3 – …
    3. –1 + 2z – 2z2 + 2z3 – …
    4. 1 – z + z2z3 + …

    Ans: (b)

  4. The region of convergence of the power series
    1. |z – 1| π 2
    2. |z – 1| π 1

    Ans: (c)

  5. Laurent’s series expansion of f (z) = z2e1/z at z = 0 is

    Ans: (c)

  6. Laurent’s series expansion of in the region 1 π |z| π 2 is

    Ans: (b)

  7. (Laurent’s). Give the regions of convergence of the two series respectively:
    1. |z| ≤ 1, |z| ≥ 1
    2. |z| π 1, |z| > 1
    3. |z| ≤ 1, |z| > 1
    4. |z| π 1, |z| ≥ 1

    Ans: (b)

  8. Laurent series of for 0 π |z| π ∞

    Ans: (b)

Zeros and Singularities

  1. z = nπ (n: integer) are the zeros of the function
    1. cos z
    2. sinh z
    3. sin z
    4. cosh z

    Ans: (c)

  2. (n: integer) are the zeros of the function
    1. cos z
    2. sinh z
    3. sin z
    4. sinh z

    Ans: (a)

  3. The singularities of coth z are the zeros of
    1. tan z
    2. sinh z
    3. coth z
    4. cosh z

    Ans: (b)

  4. The singularities of ez are the zeros of
    1. ez
    2. ez
    3. log z
    4. tan z

    Ans: (a)

  5. The singularity of ez/z2(1 – z)3 at z = 0 is a/an
    1. simple pole
    2. pole of order two
    3. essential singularity
    4. non-isolated singularity

    Ans: (b)

  6. The function has at z = 0 a/an
    1. essential singularity
    2. removable singularity
    3. pole of order two
    4. a point of continuity

    Ans: (b)

  7. Among the following, the function which has simple pole at z = 0 is f (z) =
    1. (z + 1)/z(z + 2)
    2. e–z
    3. sin z/cos z
    4. (z2 – 1)/z2(1 + z2)

    Ans: (a)

  8. Among the following, the function which has a removable singularity at z = 0 is
    1. (1 – cos z)/z

    Ans: (c)

  9. Among the following, the function which has an essential singularity at z = ∞ is
    1. ez

    Ans: (c)

  10. Among the following, the function which has a pole of order 3 at z = ∞ is
    1. sin z
    2. z + ez
    3. z2 + 2ez
    4. z3

    Ans: (d)

  11. If the principal part of Laurent’s expansion of f (z) contains no term, then the singularity z = a of f (z) is called ______ singularity.
    1. essential
    2. isolated
    3. removable
    4. non-essential

    Ans: (c)

  12. The singularity of f (z) = (sin z – cos z)–1 is at z =

    Ans: (d)

  13. The simple poles of f (z) = (tan z)/z, which lie inside the circle |z| = 2, are

    Ans: (c)

  14. The number of singularities of f (z) = ez is
    1. one
    2. two
    3. infinity
    4. zero

    Ans: (d)

  15. The simple poles of the function f (z) = (z + 1)/z3(z2 + 1) are
    1. 0, –1
    2. ±i
    3. –1, ±i
    4. ±1

    Ans: (b)

  16. The number of simple poles of f (z) = z4/(1 + z4) is
    1. 1
    2. 2
    3. 3
    4. 4

    Ans: (d)

  17. The zeros of are at z =
    1. (nz)
    2. ± (n = 1, 2, 3, …)
    3. (n = 1, 3, 5, …)
    4. ± (n = 1, 3, 5, …)

    Ans: (b)

  18. z = 0 is a zero of
    1. sin z
    2. cos z

    Ans: (c)

  19. The function f (z) = zk sin z has a zero of second order if k =
    1. 0
    2. 1
    3. 2
    4. –1

    Ans: (b)

    Hint: for k = 1, f (0) = 0,

    f ′(0) = (sin z + 2 cos z)z=0 = 0,

    f ″(0) = (2 cos z − sin z)z=0 ≠ 0

  20. The function has a pole of order at z = 0
    1. one
    2. two
    3. three
    4. four

    Ans: (b)

    Hint:

  21. The type of singularity that f (z) = (1 –cos z)/z has at z = 0 is a/an ____ singularity.
    1. essential
    2. irremovable
    3. removable
    4. non-isolated

    Ans: (c)

  22. The zeros of are at z =
    1. ()–1 (nZ))
    2. (2n + 1)π (nZ)
    3. ()–1 (n = 1, 2, 3, …)
    4. ±()–1 (n = 1, 2, 3, …)

    Ans: (d)

  23. The singularity of is at z =
    1. –2
    2. 2
    3. 3
    4. 0

    Ans: (a)

  24. The type of singularity at z = 3 for f (z) = (z + 2) sin[(z – 3)–1] is ______ singularity.
    1. removable
    2. essential
    3. pole of order m
    4. non isolated

    Ans: (b)

  25. The type of singularity at z = 0 for f (z) = (z – sin z)/z2 is ____ singularity.
    1. removable
    2. essential
    3. pole of order m
    4. non isolated

    Ans: (b)

  26. The circle inside which all the zeros of lie is |z| =

    Ans: (d)

  27. The circle inside which all the zeros of lie is |z| =

    Ans: (b)

  28. z = 0 is a removable singularity for the function f (z) =

    Ans: (c)

  29. z = 0 is an essential singularity for the function f (z) =

    Ans: (a)

  30. Among the following, the function which has infinite number of isolated singularities is
    1. ez/(z2 + 1)
    2. (sin z)–1
    3. ez

    Ans: (b)

Chapter 6 Calculus of Residues

  1.  

    1. 2
    2. 0
    3. 1
    4. 3

    Ans: (c)

  2. If z = a is a simple pole of f (z), then Res [f (z) : a] =
    1. [(za)2f (z)]z=a

    Ans: (d)

  3. If z = a is a simple pole of where p(z) is analytic at a and p(a) ≠ 0 and q(a) ≠ 0, then Res {f (z) : a} =
    1. p′(a)/q(a)
    2. p(a)/q(a)
    3. p(a)/q′(a)
    4. p′(a)/q′(a)

    Ans: (c)

  4. If z = a is a pole of order m > 1 of where g (z) is analytic at a and g (a) ≠ 0, then Res {f (z) : a}=
    1. g(m)(a)/(m − 1)!
    2. g(m)(a)/m!
    3. g(m−1)(a)/(m – 1)!
    4. g(m−1)(a)

    Ans: (c)

  5. Res {(z + 1)/(z2 – 2z) : 0} =
    1. −1
    2. 1

    Ans: (a)

  6. If f (z) = (1 + ez) / (z cosz + sinz), then Res {f (z) : 0} =
    1. 1
    2. 0
    3. −1
    4. 2

    Ans: (a)

  7. The residue of cot z at z = 0 is
    1. 0
    2. 1
    3. −1
    4. 2

    Ans: (b)

  8. If then Res {f(z) : 1} =
    1. 2
    2. ez
    3. 1
    4. 2e2

    Ans: (d)

    Hint:

     

     

  9. If , then Res {f (z) : πi} =
    1. 1

    Ans: (c)

  10. If , then a =
    1. πi
    2. −πi

    Ans: (d)

    Hint:

  11. If then C is a simple closed path such that C
    1. C1 : |z| = .5
    2. C2 : |z| = 1.5
    3. C3 : |z| = 2
    4. C2 U C3

    Ans: (d)

  12. If C is any simple closed path described in counter-clockwise direction such that 0 and 1 lie inside C, then
    1. 2πi
    2. −2πi
    3. −6πi
    4. −8πi

    Ans: (c)

  13. If C is any positively-oriented single closed path such that 0 is inside and 1 is outside, then
    1. 2πi
    2. −2πi
    3. −6πi
    4. −8πi

    Ans: (d)

  14. If C is any positively-oriented simple closed path such that 1 is inside and 0 is outside, then
    1. 2πi
    2. −2πi
    3. −6πi
    4. −8πi

    Ans: (a)

  15. The value of where C : |z| = 3/2 is
    1. tan1
    2. 2 tan1
    3. 2πi tan1
    4. 0

    Ans: (c)

    Hint:

  16. Resz = 0(zeπ/z) =
    1. π2
    2. π2/2
    3. 3π2/2
    4. 0

    Ans: (b)

    Hint:

  17. Resz = 2i[zeπz/(z2 − 16)] =

    Ans: (c)

    Hint:

  18. The value of dz where C is a positively-oriented simple closed curve enclosing z = ±2i only is
    1. πi/4
    2. −πi/4
    3. −πi/8
    4. −πi/16

    Ans: (b)

  19. The value of where C is any simple closed path is
    1. 0
    2. i
    3. i
    4. i

    Ans: (a)

  20. Resz=−2[z2/(z − 1)(z + 2)2] =
    1. 1
    2. 0
    3. 8/9
    4. 4/9

    Ans: (c)

    Hint:

  21. Resz=2[z2/(z − 1)(z − 2)2] =
    1. 1
    2. 0
    3. 4
    4. 2

    Ans: (b)

    Hint:

  22. For evaluating an integral of the type the contour to be used is
    1. the semi circle s : z = Re(0 ≤ θ ≤ π) in the upper half-plane with the line segment L[−R, R] along the real axis
    2. the rectangle with vertices at ±|a|±|b|i
    3. the sector z = Re (0 ≤ θ ≤ α)
    4. the unit circle |z| = 1

    Ans: (d)

  23. For evaluating an integral of the type the contour to be used is
    1. the semicircle s : z = Re (0 ≤ θ ≤ π) in the upper half-plane with the line segment L[-R, R] along the real axis
    2. the rectangle with vertices at ±|a| ± |b|i
    3. the sector s = z = Re(0 ≤ θ ≤ π)
    4. the unit circle |z| = 1

    Ans: (a)

  24. If C is the circle |z| = π described in counter-clockwise direction, then =
    1. πi
    2. 2πi
    3. 0
    4. −1

    Ans: (b)

  25. If C is the circle |z| = , then
    1. πi
    2. 2πi
    3. 0
    4. −1

    Ans: (c)

  26. If C is the circle |z| = 1, then
    1. πi
    2. 2πi
    3. 0
    4. −1

    Ans: (b)

  27. If C is the circle |z| = 3, then
    1. πi(e4e2)
    2. 2πi(e4e2)
    3. 0
    4. −1

    Ans: (b)

    Hint:

  28. Let f(α) If |z| = 2,α = 1, then F(1) =
    1. 4πi
    2. 2πi
    3. 0
    4. 6πi

    Ans: (a)

  29. If |z| = , α = 1, then F(1) in Qn 28 is
    1. 4πi
    2. 2πi
    3. 0
    4. 6πi

    Ans: (c)

  30. For any α lying inside C (counter-clockwise), we have F(α) =
    1. −2πi(α2 + 1)
    2. 2πiα, 2πi
    3. 0
    4. πi2 + 1)

    Ans: (b)

  31. F′(α) and F"(α) in Qn 30 are
    1. 4πiα, 4πi
    2. 2πiα, 2πi
    3. 0
    4. πi2 + 1)

    Ans: (a)

Chapter 7 Argument Principle and Rouche’s Theorem

  1. A function f (z) whose only singularities are poles is called ______ function
    1. isomorphic
    2. meromorphic
    3. endomorphic
    4. epimorphic

    Ans: (b)

  2. Among the following, the function which is meromorphic is
    1. ez
    2. sin z
    3. tan z
    4. cos z

    Ans: (c)

  3. Let f, gH(D) where D contains a closed curve C. If _____ on C, then f (z) and f (z) + g(z) have the same number of zeros inside C.
    1. |g (z)| π f (z)|
    2. |g (z)| = f (z)|
    3. |g (z)| > f (z)|
    4. f (z)||g(z)| > 1

    Ans: (a)

  4. Let f, gH(D) where D contains a closed curve C. If |g(z)| π |f (z)| on C, then f (z) and ________ have the same number of zeros inside C.
    1. g(z)
    2. |g (z)| + |f (z)|
    3. |g (z)| – |f (z)|
    4. |g (z)||f (z)|

    Ans: (b)

  5. The name of the theorem stated in Qn 4 is ________ theorem.
    1. Rouche’s
    2. Argument
    3. Liouville
    4. Fundamental theorem of Algebra

    Ans: (a)

  6. The number of roots of z6 + 16z + 1 = 0 that lie inside |z| = 2 is
    1. 2
    2. 4
    3. 5
    4. 6

    Ans: (d)

    Hint: f(z) = z6 (has 6 zeros) g(z) = 16z + 1 z6 + 16z 1 has the same no. of zeros as f, i.e., 6.

     

  7. The number of zeros of z6 + z5 + 1 inside C : |z| = 2 is
    1. 4
    2. 3
    3. 6
    4. 0

    Ans: (c)

    Hint: f(z) = z6 (six zeros) g(z) = z5 + 1 z6 + z5 + 1 has the same no. of zeros as f, i.e., 6.

     

  8. The number of zeros of z6 + z5 + 1 inside C : |z| = 2 is
    1. 4
    2. 3
    3. 6
    4. 0

    Ans: (b)

    Hint: f(z) = 2z5 + 1, g(z) = : z6 + 5z5 + 1 has the same no. of zeros as f, i.e., 5

     

  9. The number of zeros of zn + z5 + 1 inside C : |z| = 1 is
    1. 0
    2. 1
    3. 2
    4. n

    Ans: (d)

    Hint: f(z) = zn (it as n zeros); g(z) = z + 1 has the same no. of zeros as f(z), i.e., n.

     

  10. The number of zeros of zn + z5 + 1 inside C : |z| = 1 is
    1. 0
    2. 1
    3. 2
    4. n

    Ans: (b)

    Hint: f(z) = z + 1 (are zero) g(z) = zn has the same no. of zeros as f(z), i.e., 1.

     

  11. The number of zeros of zn +z+1 in the annulus 1 ≤ |z| ≥ 2 is
    1. 2
    2. n
    3. 3
    4. (n−1)

    Ans: (d)

  12. The number of zeros of 3zne that lie inside |z| = 1 is
    1. 1
    2. 2
    3. n
    4. n−1

    Ans: (c)

  13. All the roots of πz2ez + 0.2 lie inside the circle |z| =
    1. 0.1
    2. 0.25
    3. 0.5
    4. 1

    Ans: (d)

    Hint:

     

  14. z7−5z3 + 12z2 has ________ zeros in the annulus 1 ≤ |z| ≤ 2.
    1. 7
    2. 6
    3. 5
    4. 0

    Ans: (c)

    Hint:

  15. The number of roots of z7 − 5z3 = 0 that lie in the punctured disc 0 π |z| π 3/2 is
    1. 7
    2. 4
    3. 3
    4. 0

    Ans: (b)

    Hint:

Chapter 8 Conformal Mapping

  1. A mapping that preserves angles between oriented curves both in magnitude and in sense is called a/an ________ mapping.
    1. informal
    2. isogonal
    3. conformal
    4. formal

    Ans: (c)

  2. The mapping defined by an analytic function f (z) is conformal at all points z except at points where ______
    1. f’(z) = 0
    2. f’(z) ≠ 0
    3. f’(z) > 0
    4. f’(z) π 0

    Ans: (a)

  3. Under the mapping w = z2, a circle |z| = c is transformed to a/an
    1. ellipse a2u2 + b2v2 = 1
    2. circle |w| = c2
    3. vertical line
    4. horizontal line

    Ans: (b)

  4. The mapping w = z2 transforms a vertical line to a/an
    1. horizontal line
    2. circle
    3. ellipse
    4. parabola

    Ans: (d)

  5. Under the mapping w = f (z) if the angle θ = π/n is transformed to an angle π in the w-plane, then f (z)
    1. zn
    2. z2n
    3. z1/n
    4. z/n

    Ans: (a)

  6. Under the transformation w = iz, the st. line y = x in the z-plane is rotated through an angle _______in the w-plane.
    1. π/4
    2. 2π
    3. π/2
    4. π

    Ans: (c)

  7. The mapping w = preserves angles in
    1. size but not in sense
    2. sense but not in size
    3. size as well as sense
    4. neither size nor sense

    Ans: (a)

  8. The mapping w = |z|2 is not conformal at
    1. any point
    2. any point except z = 0
    3. the origin
    4. any point along the real axis

    Ans: (b)

  9. ez maps
    1. vertical lines into circles
    2. circles into vertical lines
    3. circles into circles
    4. straight lines into straight lines

    Ans: (a)

  10. ez maps
    1. circles into circles
    2. circles into horizontal lines
    3. horizontal lines into rays through O
    4. rays through O into circles

    Ans: (c)

  11. The two points where the mapping is not conformal are z =
    1. 0, 1
    2. ±∞
    3. 0, 0
    4. ±1

    Ans: (d)

  12. The images of the vertical lines under the mapping w = sin z are
    1. hyperbolas
    2. ellipes
    3. circles
    4. parabolas

    Ans: (a)

  13. The images of the horizontal lines under the mapping w = sin z are
    1. hyperbolas
    2. ellipes
    3. circles
    4. parabolas

    Ans: (b)

  14. The condition for the map w = (az + b)/(cz + d) to be conformal for all z is that

    Ans: (c)

  15. The transformation w = (az + b)/(cz + d) maps the unit circle |w| = 1 into a st. line in the z-plane if
    1. |a| = |c|
    2. |a| = |d|
    3. |b| = |c|
    4. |b| = |d|

    Ans: (a)

  16. The fixed points of the transformation w = z2 are
    1. 0, 1
    2. 0, −1
    3. −1, 1
    4. i, i

    Ans: (a)

  17. The invariant points of the mapping w = are
    1. 1, −1
    2. 0, −1
    3. 0, 1
    4. −1, −1

    Ans: (c)

  18. The fixed points of w = are
    1. ±1
    2. ±i
    3. 0, −1
    4. 0, 1

    Ans: (b)

  19. The mapping w = transforms circles of constant radius into
    1. confocal ellipses
    2. hyperbolas
    3. circles
    4. parabolas

    Ans: (a)

  20. Among the following, the one that is not a critical point of the mapping w = cos z is
    1. π
    2. 2π
    3. 3π
    4. π/2

    Ans: (d)

  21. The linear fractional transformation that maps the points ∞, 1, 0, respectively, into 0, 1, ∞ is w =
    1. z
    2. z−1
    3. iz
    4. iz

    Ans: (b)

  22. The mobius transformation that maps the points 0, i, ∞, respectively, into 0, 1, ∞ is w =
    1. z−1
    2. −z
    3. iz
    4. iz

    Ans: (c)

  23. The images of the points z1 = ∞, z2 = i, z3 = 0 under the transformation w = −1 / z are
    1. 0, i, ∞
    2. 0, − i, ∞
    3. 0, −i, −∞
    4. 0, −i, 1

    Ans: (c)

  24. The fixed points of the transformation w = are
    1. 1, −1, −i
    2. i, 1, −i
    3. 1, −1, i
    4. i, −1, −i

    Ans: (d)

  25. Under the transformation w = , the point is mapped into the point
    1. (cosπ, sinπ)

    Ans: (b)

  26. The image of the line x = 1 under the transformation w = sin z is a/an
    1. circle
    2. ellipse
    3. hyperbola
    4. parabola

    Ans: (c)

  27. The image of the line y = c under the transformation w = sin z is a/an
    1. circle
    2. ellipse
    3. hyperbola
    4. parabola

    Ans: (b)

  28. The mapping under which a vertical line z = c transforms into a circle is w =
    1. sin z
    2. cosh z
    3. z2
    4. ez

    Ans: (d)

  29. z = 0 is a critical point of the transformation w =
    1. sin z
    2. zn(n > 1)
    3. ez

    Ans: (b)

  30. The mapping under which vertical lines are transformed into a circle is w =
    1. z2
    2. ez
    3. sin z

    Ans: (c) z

  31. Under the mapping w = z2, a st. line through the origin is transformed into a/an
    1. circle
    2. ellipse
    3. parabola
    4. st. line through O

    Ans: (d)

  32. Under the mapping w = z2, a circle of radius 3 is transformed into a circle with radius
    1. √3
    2. 3
    3. 9
    4. 6

    Ans: (c)

  33. Under the mapping, the circle of radius ________ is transformed into a circle with radius .
    1. 1

    Ans: (c)

  34. The image of the circles | w − 1| = k under the mapping w = z2 is given by
    1. |z − 1||z + 1|= k
    2. |z − 1|= k
    3. |z + 1|= k
    4. z2 = k

    Ans: (a)

  35. The fixed points of the transformation w = z/(z + 1) are
    1. 0, −1
    2. 0, 1
    3. 1, −1
    4. 2, 1

    Ans: (a)

  36. The sector 0 ≤ 0 ≤ is mapped onto the upper half−plane by w = zn if n =
    1. 3
    2. 6
    3. 3
    4. 4

    Ans: (b)

  37. The number of fixed points of the mapping w = is
    1. 1
    2. 2
    3. 3
    4. 4

    Ans: (b)

  38. The equation of the line whose image under the map w = ez is the circle with radius with radius ‘e’ is
    1. x = 0
    2. y = 1
    3. y = 0
    4. x = 1

    Ans: (d)

  39. The ______ mapping has every point of it as a fixed point.
    1. rotation
    2. inverse
    3. identity
    4. traslation

    Ans: (c)

  40. w = has ________ fixed points.
    1. no
    2. infinitely many
    3. finitely many
    4. some

    Ans: (b)

  41. The translation mapping w = z + c (c ≠ 0) has _________ fixed points.
    1. no
    2. infinitely many
    3. finitely many
    4. some

    Ans: (a)

  42. The rotation mapping w = cz has ________ fixed point(s).
    1. no
    2. infinitely many
    3. finitely many
    4. one

    Ans: (d)

    Hint: z = cz =)(c − 1)z = 0 ⇒ z = 0

Fill in the blanks

Chapter I Special Functions

A. Beta and Gamma Functions

  1. _______.

    Ans:

  2. _______.

    Ans: 3/2

  3. _______.

    Ans: 2

  4. _______.

    Ans:

  5. _______.

    Ans:

  6. _______.

    Ans: π cosec

  7. B(1/3, 2/3) = _______.

     

    Ans:

  8. B(1/2, 1/2) = _______.

    Ans: π

  9. _______.

    Ans: –2

  10. _______.

    Ans: 1/2

  11. B(1, 1/2) = _______.

    Ans: 2

  12. Γ(3/2)Γ(1/2) = _______.

    Ans: π/2

  13. B(p + 1, q) + B(p, q + 1) = _______.

    Ans: (p, q)

  14. B(p + 1, q)/B(p, q) = _______.

    Ans: p/p + q

  15. Γ(1/2) = _______.

     

    Ans:

  16. Γ(3/2) = _______.

    Ans:

  17. Γ(–1/2) = _______.

     

    Ans:

  18. Γ(x) = xx = _______.

     

    Ans: 1

  19. Γ(–3/4) = _______ Γ(1/4).

     

    Ans: –4/3

  20. Γ(1/4)Γ(3/4) = _______.

     

    Ans:

  21. Γ(1/3)Γ(2/3) = _______.

     

    Ans:

  22. _______.

    Ans: 1/3

  23. _______.

    Ans:

  24. _______.

    Ans: 22p–1

  25. Γ(p)Γ(1 – p) = πp = _______.

    Ans: 1/2

  26. _______.

    Ans: Γ(n)/an

  27. _______.

    Ans: 2

  28. _______.

    Ans: Γ(1/p + 1)

  29. If tan α = 1 (0 π α π π/2) and Γ2(x) = α then x = _______.

     

    Ans: 3/2

  30. _______.

    Ans:

  31. _______.

    Ans: π

  32. (m + n) B (m + 1, n) = _______.

     

    Ans: mB(m, n)

  33. _______.

    Ans: π

  34. _______.

    Ans:

  35. _______.

    Ans: (1/2, 1/2)

  36. _______.

    Ans: B(2, 3)

  37. _______.

    Ans:

  38. _______.

    Ans: B(5, 6)

B. Legendre Functions

  1. [P1(–x)]1 = ________.

     

    Ans: –1

  2. ________.

    Ans: 2/(2n + 1)

  3. Rodringue’s Formula is Pn(x) = CDn[(x2 – 1)n] with C = ________.

     

    Ans: 1/(2nn!)

  4. The generating function for Pn(x) is ________.

     

    Ans: (1 – 2xt + t2)–1/2

  5. The sum of the series 1 + t + t2 + t3 + … in terms of Legendre function is ________.

     

    Ans: Pn (1)

  6. P0(x) = ________.

     

    Ans: 1

  7. ________ (mn).

    Ans: 0

  8. P11(x) = ________.

     

    Ans: P0 (x)

  9. Pn(1) = ________.

     

    Ans: 1

  10. Pn(–x) = ________ Pn(x).

     

    Ans: (– 1)n

  11. Pn(– 1) = _________

     

    Ans: (– 1)n

  12. ________.

     

    Ans: 2/7

  13. ________.

     

    Ans: 0

  14. If x2 + x = aP0 + bP1 + cP2, then (a, b, c) = ________.

    Ans:

  15. P2n(0) = _______

     

    Ans: (–1)n(2n)!/22n(n!)2

  16. ________.

    Ans:

  17. Legendre’s differential equation is ________.

     

    Ans: [(1 – x2)y′]′ + x(x + 1)y = 0

  18. (n + 1)Pn+1(x) + nPn–1(x) = ________.

     

    Ans: (2n + 1)xPn(x)

  19. Pn+1 (x) – Pn – 1(x) = ________.

     

    Ans: (2n + 1)Pn(x)

  20. ________.

    Ans: 2n(n + 1)/(2n + 1)

C. Bessel Functions

  1. Physical and Engineering Problems involving vibrations or heat conduction in cylindrical regions give rise to _________ equation.

     

    Ans: Bessel’s differential

  2. Bessel’s differential equation of order n is ________.

    Ans:

  3. Solutions of Bessel’s differential equation are ________called of order n.

     

    Ans: Bessel’s functions

  4. The series expansion of Jn (x) = ________.

    Ans:

  5. Jn(x) is called _________.

     

    Ans: Bessel function of the first kind of order n

  6. When n is an integer a second linearly independent solution of Bessel’s differential equation is ________.

    Ans:

  7. _________ is called the Bessel function of the second kind of order n or Neumann function.

     

    Ans: ym(x)

  8. J1/2(x) = ________.

     

    Ans:

  9. J–1/2(x) = ________.

    Ans:

  10. ________.

     

    Ans: Jn+1 (x)

  11. The generating function for Jn (x) is ________.

    Ans:

  12. ________.

     

    Ans: Jn(x)

  13. ________.

    Ans: J2m (x)

  14. J20 + 2(J21 + J22 + …) = ________.

     

    Ans: 1

Chapter 2 Functions of a Complex Variable

  1. f (z) = is ________.

     

    Ans: not analytic for any z

  2. The Cauchy−Riemann equation is only ________ but not ________ for analytic of f (z)

     

    Ans: necessary, sufficient

  3. f (z) = |z|2 is ________.

     

    Ans: continuous for all z but derivable at z = 0 only

  4. is ________.

    Ans: analytic for all finite z ≠ 1

  5. The area of the triangle formed by the points 1 + i, i – 1, 2i (in sq. units) is ________.

     

    Ans: 1

  6. The triangle formed by joining the points (±1 +) and the origin is ________.

     

    Ans: equilateral

  7. The discontinuity of at z = −i may be removed by defining f (–i) = ________.

     

    Ans: i

  8. An analytic function of constant absolute value is ________.

     

    Ans: constant

  9. A function u(x, y) having continuous second partial derivatives and satisfying ∇2u = 0 is called a ________ function.

     

    Ans: harmonic

  10. The analytic function whose imaginary part is v(x, y) = 2xy is ________.

     

    Ans: z2

  11. The harmonic conjugate of v(x, y) = 2xy is ________.

     

    Ans: x2y2

  12. If f(z) = z/|z| (z ≠ 0) and f(0) = 0, then f is ________

     

    Ans: discontinuous at z = 0 only

  13. If u(x, y) = 2xx2+my2 is harmonic in D, then m = ________.

     

    Ans: 1

  14. If f (z) = z , then f ’(z) exists ________.

     

    Ans: z = 0 only

  15. If f (z) = u + iv is analytic, then ________.

     

    Ans: Ux = Vy, Uy = – Vx

  16. f (z) = (1 + z)/(1 – z) is differentiable at ________.

     

    Ans: all z except z = 1

  17. f (z) = Re(z) is differentiable at ________.

     

    Ans: no z

  18. The curves u(x, y) = ex cos y = c′ are orthogonal to the curves v(x, y) = c2 where v(x, y) = ________.

     

    Ans: ex sin y

Chapter 3 Elementary Functions

  1. sin(x + iy) = ________.

     

    Ans: (sin x cosh y + i cos x sinh y)

  2. cos(x + iy) = ________.

     

    Ans: (cos x cosh yi sin x sinh y)

  3. Re[tan(x + iy)] = ________.

     

    Ans: sin2x/(cos2x + cosh y)

  4. | cos(x + iy)|2 = ________.

     

    Ans: (cos2 x + sin2 y)

  5. Sin z is periodic with period ________.

     

    Ans: 2

  6. Tan z is periodic with period ________.

     

    Ans: π

  7. ez is periodic with period ________.

     

    Ans: 2πi

  8. ________.

     

    Ans:

  9. ________.

     

    Ans:

  10. The solutions of cos z = 0 are z = ________.

    Ans: (2n + 1)

  11. When expressed in terms of natural logarithm, sin–1 z = ________.

     

    Ans:

  12. When expressed in terms of natural logarithm, cos–1 z = ________.

     

    Ans:

  13. All the roots of cos z = 2 are given by z = ________.

     

    Ans:

  14. All the roots of sinh z = i are given by z = ________.

     

    Ans:

  15. When expressed in terms of logarithm, tan–1 z = ________.

  16. When expressed in terms of logarithm, sec–1 z = ________.

    Ans:

Chapter 4 Complex Integration

  1. ________.

    Ans: 1 – i

  2. where C is the upper half of the unit circle taken in the clockwise direction, is ________.

     

    Ans:πi

  3. If C is the circle |za| = r, then ________.

    Ans: 2πi

  4. If C is the circle |za| = r and n ≠ –1, then ________.

    Ans: 0

  5. If C is the straight path y = x from z = 0 to 1 + i, then ________.

    Ans:

  6. ________.

    Ans:

  7. ________.

    Ans:

  8. ________.

    Ans:

  9. If C is the arc of the circle |z| = 2 from θ = 0 to θ = π/3, then ________.

    Ans:

  10. If C is the left half of the unit circle in the clockwise direction, then ________.

    Ans: 2i

  11. By the ML-inequality, an upper bound for is ________.

    Ans:

  12. By the Cauchy’s Integral Theorem, ________ (C : |z| = 1).

     

    Ans: 0

  13. By the Cauchy’s Integral Theorem, ________ (C : |z| = 1).

     

    Ans: 0

  14. By the Extended Cauchy Integral Theorem for multiply-connected domains, (C : |z| = 2) = ________.

    Ans: 4πi

  15. By the Cauchy’s Integral Theorem, (C : |z – i| = 2) = ________ (C : |z| = 1).

    Ans:

  16. If C is the circle |z| = 2, ________.

    Ans:

Chapter 5 Complex Power Series

  1. For |z| > 1, (1 + z)–1 = ________.

    Ans:

  2. Taylor’s series about z = 0 of is ________.

    Ans:

  3. The region of convergence of the series in Qn. 2 is ________.

     

    Ans: R : |z| π 1

  4. Laurent’s series expansion about z = 1 of is ________.

    Ans:

  5. The region of convergence of the series in Qn. 4 is ________.

     

    Ans: R : 0 π |z – 1| π 2

  6. Taylor’s series for about z = 0 is ________.

    Ans:

  7. The region of convergence of the series in Qn. 6 is ________.

     

    Ans: R : 0 π |z| π 1

  8. Laurent’s series expansion for |z| > 1 of f (z) = is ________.

    Ans:

  9. Laurent’s series expansion for |z – 1| > 1 of is ________.

    Ans:

  10. Laurent’s series expansion for |z| > 2 of f (Z) = is ________.

    Ans:

Chapter 6 Calculus of Residues

  1. The poles of are z = ________.

    Ans: ±i

  2. The Residues of is ________.

    Ans:

  3. The Residues of is ________.

    Ans:

  4. If C is the circle |z| = 2, then by the Residue Theorem ________.

    Ans: 2πi

  5. The pole of ________.

    Ans: 1

  6. The order of the pole in Qn. 5 is ________.

     

    Ans: 3

  7. The Residue at z = 1 of f (z) = is ________.

    Ans:

  8. By the Residue Theorem, where C : |z| = 2 is ________.

    Ans: 3πie

  9. The poles of are z = ________.

    Ans: 1, 2

  10. The Residue of f(z) at z = 1 in Qn. 9 is________ .

     

    Ans: 1

  11. If C : |z| = 3/2, then ________.

    Ans: 2πi

  12. If C: |z| = 5, then ________.

    Ans: 2πi cos 3

  13. ________.

    Ans:

  14. ________.

    Ans: π/12

  15. ________.

    Ans: π/6

Chapter 7 Argument Principle And Rouche’S Theorem

  1. A complex function f (z) which is differentiable for all z in a domain D is called a/an ________ in D.

     

    Ans: analytic or holomorphic or regular

  2. A point z = a where f(z) is not differentiable is called a/an ________ of f(z).

     

    Ans: singularity

  3. A point z = a at which f(z) is not analytic but there exists a nbd of ‘a’ such that at each point of it except z = a, f(z) is analytic is called a/an ________ of ‘a’.

     

    Ans: isolated singularity

  4. If z = a is an isolated singularity for f (z) such that (za)mf (z) = ϕ(D) ∈ H(D) for some positive integer m and ϕ(a) ≠ 0 is called a ________.

     

    Ans: pole of order m

  5. If m = 1 in Qn. 4, then it is called a ________.

     

    Ans: simple pole

  6. A complex function in a domain D whose singularities are only poles is called ________function.

     

    Ans: meromorphic

  7. The change in the number of radius in the argument of f(z) as the point z makes a cycle in the positive direction is given by ________.

    Ans:

  8. By Argument Principle, if f (z) is meromorphic in a domain D and C is any simple closed curve in D, then ________.

    Ans:[N – P]

  9. Rouche’s theorem states that if f, gH(D) and C is a closed curve in D and |g(z)|π f (z)| on C then ________ and ________ have the same number of zeros in C.

     

    Ans: f(z), f(z) + g(z)

  10. Let D be a bounded domain f (z) ∈ H (D) and nonconstant in D. Then max |f(z)| occurs on the boundary of D. This is the statement of the ________ Theorem.

     

    Ans: Maximum modulus

Chapter 8 Conformal Mapping

  1. A mapping that preserves the angles between oriented curves in magnitude and sense is called a ________.

     

    Ans: conformal mapping

  2. The points which are mapped onto themselves under a conformal mapping are called ________ points.

     

    Ans: fixed invariant

  3. The fixed points of = ________.

    Ans: [0, 1]

  4. The fixed points of z = ________.

    Ans: ±i

  5. The fixed points of _______.

    Ans: [3, 3]

  6. The fixed points of z = ________.

    Ans: – 1 ± 2i

  7. The critical points of the mapping w = z2 is z = ________.

     

    Ans: 0

  8. The critical points of the mapping w = z + ________.

    Ans: ±1

  9. The invariant points of the transformation w = are Z = ________.

    Ans: ±i

  10. The invariant points of the transformation w = are Z = ________.

    Ans: 3i ± 2

  11. The transformation w = sin z maps st. lines x = c into the family of ________.

     

    Ans: confocal hyperbolas

  12. For the transformation w = f(z), the point where f’ (z) = 0 is called a ________ point.

     

    Ans: critical

  13. The transformation w = cosh z maps horizontal lines y = c into the family of ________.

     

    Ans: confocal hyperbolas

  14. The transformation w = cosh z is not conformal at points where z = ________.

     

    Ans: nπi

  15. The image of |z – 1| = 1 under the mapping ________.

    Ans:

  16. Under the bilinear transformation maps circles into ________ or ________.

     

    Ans: circles, st. lines

  17. Under the transformation w = z2, the image of the line z = 4 is a ________ whose equation is ________.

     

    Ans: parabola, v2 = –64(u –16)

  18. The Linear Fractional Transformation that maps –1, i, 1 into 0, 1, ∞ is w = ________.

     

    Ans: (z + 1)/(z – 1)

Match the Following

Chapter 1 Special Functions

A. Beta and Gamma Functions

    A

B

    1. B (p, q)

a. Γ(p + 1)

    2. p Γ (p)

b. π sin π/4

    3. Γ (p) Γ(1. p)

c. 22p–1 Γ (p) Γ(p + 1/2)

    4.


d. π cosec

    5. Γ(2p)

e. Γ(p)Γ(q)/Γ(p + q)

 

f. π sin

Ans:    1. c    2. a    3. d    4. b    5. c

B. Legendre Functions

    A (Function)

B (Value)

    1. P2n+1(0)

a. (–1)n(2n)!/22n(n!)2

    2. P2n(0)

b. (–1)n


    3. Pn(−x)/Pn(x)

c.

    4.

d.

    5. Pm(x)Pn(x)dx


e. 1 + 2 + 3 + … + n

 

f. 0

Ans:    1. f    2. a    3. b    4. e    5. c

C. Bessel Functions

    A

B

    1.

a.


    2. (xJnJn+1)

b.

    3.

c.

    4.

d. 2/πx

    5.

e. (2/pix)2

 

f. 1

Ans:    1. c    2. a    3. b    4. f    5. d

Chapter 2 Functions of a Complex Variable

A

B


1. ∇2u = 0

a.


2. f(z) = u(x, y) + iv(x, y) ∈ H(D)

b.

3. u(x, y) = 2 log(x2 + y2) + k

c. ux = vy;   ux = –vy

4. u(r, θ) = r2 cos 2θ = c

d. u is a harmonic function

5.

e. v(r, θ) = r2 sin 2θ = k

 

f.

Ans:    1. d    2. c    3. f    4. e    5. b

Chapter 3 Elementary Functions

    A (Function/Equation)

B (General value)


    1. sin z = cosh 4

a.


    2. log(1 + i)

b.

    3. (1 + i)i

c.


    4. ii

d.

    5. sin –1 z

e.

 

f.

Ans:    1. d    2. c    3. a    4. b    5. f

Chapter 4 Complex Integration

  A (Integral)

B (Value)

  1.


a. 0

  2.


b. –π

  3.


c. 2πi/9

  4.


d. π/8

  5.


e. 2πi

 

f. πi

Ans:    1. c    2. a    3. d    4. b    5. f

Chapter 5 Complex Power Series

  A (Integral)

B (Type of Singularity)

  1.


a. Removable singularity

  2.


b. Simple pole

  3.


c. Infinite number of isolated singularities

  4. f(z) = ez

d. Pole of higher order

  5.


e. Essential singularities at z = ∞

 

f. Finite number of isolated singularities

Ans:    1. f    2. c    3. a    4. e    5. b

Chapter 6 Calculus of Residues

   A (Function/Point)

B (Residue)

   1.

a.

   2.


b. –4/3

   3.


c. –1

   4. Res z= /2 tan z


d. 4/9

   5.


e. (10i)–1

 

f.

Ans:    1. d    2. b    3. a    4. c    5. f

Chapter 7 Argument Principle and Rouche’S Theorem

  A (Equation)

B (Location of Roots)

  1. z5 + 15z + 1 = 0

a. 1 root in each quadrant

  2. ez = az4 (a > e)

b. 1 root in the first quadrant


  3. z4 + z + 1 = 0

c.

  4. z4 + 4(1 + i) + 1 = 0

d. 4 roots lie in the first quadrant

  5. z4 + z3 + 1 = 0

e. 4 roots in |z| ≤ 1

 

f. 1 complex root in the first quadrant

Ans:    1. e    2. c    3. b    4. a    5. f

Chapter 8 Conformal Mapping

  A (Mapping of points)

B (Bilinear transformation)

  1. (–1, i, 1) → (0, i, ∞)

a. (1 + iz)/(1 – iz)

  2. (∞, i, 0) → (–1, –i, 1)

b. w = 3i(z + 1)/(3 – z)

  3. (–1, 0, 1) → (0, i, 3i)

c. w = –1/z

  4. (∞, i, 0) → (0, i, ∞)

d. w = (1 – z)/1 (1 + z)

  5. (1, i, –1) → (i, 0, –i)

e. w = –(z + 1)/(z – 1)

 

f. w = (–6z + 2i)/(iz – 3)

Ans:    1. e    2. d    3. b    4. c    5. a

 

  A (Transformation)

B (Fixed points)

  1. w = (z – 2i)–1

a. i

  2. w = (6z – 9)/z

b. ±1

  3. w = (z – 1 + i)/(z + 2)

c. (3, 3)

  4.


d. 1, 2

  5.


e. 2, 2

 

f. i

Ans:    1. f    2. c    3. a    4. b    5. d

True or False Statements

Chapter 1 Special Functions

A. Beta and Gamma Functions

  1. T

  2. F

  3. T

  4. F

  5. Γ(p)Γ(1 + p) = π cosec

     

    F

  6. T

B. Legendre Functions

  1. T

  2. Rodrigues’ Formula is

    T

  3. F

  4. if n is odd

    F

  5. Pn(1) = 1

     

    T

  6. Pn(–x) = (–1)nPn(x)

     

    T

C. Bessel Functions

  1. The differential equation satisfied by J0(x) is xy″ + y′ + xy = 0

     

    T

  2. F

  3. T

  4. T

  5. J′0(x) = J1(x)

     

    F

  6. cos x = J0 – 2J2 + 2J4

     

    T

Chapter 2 Functions of a Complex Variable

  1. f (z) = /z is not continuous at z = 0

     

    T

  2. If a complex function is derivable at a point then it is continuous there.

     

    T

  3. If a function f (z) = u + iv satisfies Cauchy–Riemann equations, then f (z) is analytic

     

    F

  4. Laplace equation in polar coordinates

    T

  5. f (x +iy) = 2xy +i(x2y2) is analytic

     

    F

  6. is analytic

    F

Chapter 3 Elementary Functions

  1. sin(iz) = i sinh z

     

    T

  2. (coth z)′ = cosech2z

     

    F

  3. Principal value of ii is eπ/2

     

    F

  4. A solution of ez = 1 + 2i is z =

    T

  5. The principal value of log(1 + i) + log(1 – i) is log 2

     

    T

  6. F

Chapter 4 Complex Integration

  1. If C is the line segment from z = 0 to z = i,

    F

  2. T

  3. If C is the unit circle |z| = 1 then

    F

  4. T

  5. is independent of path c of integration

    F

  6. If C is |z| = 3 then 8πi

    T

Chapter 5 Complex Power Series

  1. A series of the form is called a power series

     

    T

  2. If a series ∑ un (x) converges for every z in C : |za| = R, then C is called the circle of convergence

     

    T

  3. Taylor’s series expansion of a function f (z) consists of both positive and negative powers of (za)

     

    F

  4. If is expanded in the annular region 1 π |z| π 2 the series we obtain is Taylor’s series

     

    F

  5. Taylor’s series is a particular case of Laurent’s series

     

    T

  6. If f (z) has a Taylor’s series expansion valid in a region D, then f (z) ∈ H (D)

     

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Chapter 6 Calculus of Residues

  1. By Cauchy’s Residue Theorem, real integrals can be evaluated.

     

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  2. If then b1 is called the residue of f (z) at z = a

     

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  3. F

  4. F

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  6. If C is |z| = 2, then

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Chapter 7 Argument Principle and Rouche’S Theorem

  1. Liouville’s Theorem states that if f (z) ∈ H(D) and |f (z)| is bounded in D then f (z) is constant in D

     

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  2. Every polynomial of degree n has n roots

     

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  3. If f (z) ∈ H (D) then min |f (z)| or max |f (z)| can occur at any point of D

     

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  4. z5 + 15z + 1 = 0 has all its 5 roots in the unit circle |z| = 1

     

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  5. ez = zn has n roots inside the unit circle

     

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  6. One root of z4 + z3 + 1 = 0 lies in the first quadrant

     

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Chapter 8 Conformal Mapping

  1. The fixed points of the mapping is z = i

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  2. The mapping w = z1/n maps sectors into half planes

     

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  3. The image of the circle |z + 1| = 1 under the mapping is the circle

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  4. A bilinear transformation preserves the cross-ratio of four points

     

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  5. The critical points of w = cos z are at z =

     

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  6. If the mapping w = f (z) is conformal in D, then f (z) ∈ H (D)

     

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