Question Bank
Mulitple Choice Questions
Chapter 1 Special Functions
A. Beta and Gamma Functions
 If Γ(x) = 1, then x =
 1
 0
 −1/2
 1/2
Ans: (a)
 Γ(1/2) =
 π/2
Ans: (c)
 If p = 1/4, then Γ(p) Γ(1 – Γp) =
Ans: (d)

 2/3
 1/3
 3/2
 3
Ans: (b)

 π/2
Ans: (a)
 If Γ(2p) = aΓ(p) Γ(p + 1/2), then a =
 2^{2p–1}
 2^{2p+1}
 2^{2p}
 2^{p–1}
Ans: (a)
 B(m + 1, n)/B(m, n) =
 n/(m + n)
 (m + n)/m
 m/(m + n)
 (m + n)/n
Ans: (c)
 Γ(p) Γ(1 – p) = π ⇒ p =
 1/4
 2
 1/2
Ans: (d)

 Г(n)/a^{n}
 Г(n)/a
 Г(n + 1)/a^{n}
 Г(n)/a^{n−1}
Ans: (a)

 2
 1
 0
Ans: (a)

 Г(p)
 Г(1/p)
 Г(p^{2})
 Г(1 − p)
Ans: (b)

 3π/2
 π/2
 π
 0
Ans: (c)

 π
 π/2
Ans: (d)

 2π
 π/2
Ans: (b)

Ans: (a)
B. Legendre Functions

 2/(2n – 1)
 2/(2n + 1))
 1/(2n + 1)
 1/(2n – 1)
Ans: (b)
 The generating function for P_{n}(x) is
 (1 – 2xt – t^{2})^{–1/2}
 (1 – 2xt + t^{2})^{1/2}
 (1 – 2xt + t^{2})^{–1/2}
 (1 + 2xt + t^{2})^{–1/2}
Ans: (c)
 1/(1 + i) = a + bi = ⇒ (a, b) =
 (1, –1)
Ans: (a)
 P_{0} (x)
 1
 3/2
 0
 2/3
Ans: (a)

 0
 1
 –1
 1/2
Ans: (c)
 By Rodrigue’s formula, P_{n}(x) = CD^{n}{(x^{2}–1)^{n}} where the constant C =
 1/[2^{n}(2n)!]
 1/n!
 1/2^{n}
 1/(2^{n} n!)
Ans: (d)
 (1 + x) when expressed in terms of Legendre polynomials
 P_{0}(x) – P_{1}(x)
 P_{0}(x) + P_{1}(x)
 2P_{0}(x) – P_{1}(x)
 P_{0}(x) + 2P_{1}(x)
Ans: (b)
 xP′_{n–1}(x) + n P_{n–1}(x) =
 P′_{n–1}(x)
 P_{n}(x)
 P′_{n}(x)
 P_{n+1}(x)
Ans: (c)
 The degree of the polynomial P_{4}(x) is
 3
 2
 4
 1
Ans: (c)
 P_{n}(–x) = a^{n}P_{n}(x) where a =
 2
 0
 1
 –1
Ans: (d)
 (3x^{2} – 1) when expressed in Legendre polynomials
 P_{2} + 2P_{1}
 3P_{2} – P_{1}
 2P_{2}
 3P_{2} – P_{1}
Ans: (c)

 1
 2
 0
 3
Ans: (b)

 2/(2n + 1)
 0
 2/(2m + n)
 2/(m + n)
Ans: (b)
 P_{0} (x)
 1
 3/2
 0
 2/3
Ans: (a)

 0
 1
 –1
 1/2
Ans: (c)
 By Rodrigue’sformula P_{n}(x) = CD^{n}{(x^{2}–1)^{n}} where the constant C =
 1/[2^{n}(2n)!]
 1/n!
 1/2^{n}
 1/(2^{n} n!)
Ans: (d)

 2/(2n – 1)
 2/(2n + 1))
 1/(2n + 1)
 1/(2n – 1)
Ans: (b)
 The generating function of Legendre function P_{n}(x) is
 (1 – 2xt – t^{2})^{–1/2}
 (1 – 2xt + t^{2})^{1/2}
 (1 – 2xt + t^{2})^{–1/2}
 (1 + 2xt + t^{2})^{–1/2}
Ans: (c)
C. Bessel Functions

 x^{n−1}J_{n−1}
 x^{n}J_{n−1}(x)
 x^{n+1}J_{n}(x)
 x^{n−1}J_{n}(x)
Ans: (b)

 −x^{−n}J_{m}
 −x^{−n+1}J_{n}(x)
 −x^{−n}J_{n+1}(x)
 x^{−n}J_{n+1}(x)
Ans: (c)
 J–_{1/2}(x) =
Ans: (a)
 J_{n–1}(x) – J_{n+1}(x) =
 J_{n}(x)
 2J_{n}(x)
 2J′_{n}(x)
 J′_{n}(x)
Ans: (c)

 x^{n}J_{n+1}(x)
 x^{−n}J_{n−1}(x)
 x^{n}J_{n−1}(x)
 J_{n−1}(x)
Ans: (d)
 J_{n}(–x) =
 (–1)^{n}J_{n}(x)
 J_{n}(x)
 (–1)^{n–1}J_{n}(x)
 (–1)^{n}J_{n–1}(x)
Ans: (a)

Ans: (a)

 cosec x
 cot x
 cos x
 sec x
Ans: (b)
 J_{1}(0) =
 0
 1
 –1
 2
Ans: (a)
 J_{0}(0) =
 1
 0
 –1
 x
Ans: (b)
 J_{0}(x) is a solution of
 (xy′)′+ xy = 0
 (y′/x)′ + y = 0
 (x^{2}y′) + xy = 0
 x(y′/x)′ + y = 0
Ans: (a)
 J_{1/2}(x) = p(x) sin x where p(x) =
Ans: (c)
 [x^{n}J_{n}(x)]′ =
 x^{n}J_{n+1}(x)
 x^{−n}J_{n−1}(x)
 x^{n}J_{n−1}(x)
 J_{n−1}(x)
Ans: (c)

 1/(πx)
 πx/2
 2πx
 2/(πx)
Ans: (d)

 xJ_{1}(x) − J_{0}(x)
 xJ_{1}(x)
 J_{1}(x)
 x^{2}J_{n}(x)
Ans: (b)

Ans: (a)

 J_{1}(x)
 –J_{1}(x)
 (–1)^{n}J_{n}(x)
 xJ_{1}(x)
Ans: (b)
 [xJ_{1}(x)]′ =
 2J_{2}(x)
 (−1)J_{1}(x)
 xJ_{0}(x)
Ans: (c)
 J_{n}(–x) =
 J_{n}(x)
 (–1)^{n}J_{n}(x)
 2J_{n}(x)
 J_{–n}(x)
Ans: (b)

 J_{n}(x)
 J_{–n}(x)
 (–1)^{n}J_{n}(x)
 J_{n}(–x)
Ans: (a)
Chapter 2 Functions Of A Complex Variable
 f (z) = (z + i)/(z – i) is analytic
 for all z
 for all z ≠ i
 for all z ≠ – i
 for no z
Ans: (b)
 f (z) = is analytic
 at z = 0 only
 everywhere
 nowhere
 if z ≠ 0
Ans: (c)
 f (z) = z^{2} is analytic
 at z = 0 only
 everywhere
 nowhere
 if z ≠ 0
Ans: (a)
 The harmonic conjugate of u(x, y) = x^{2} – y^{2} is v =
 x^{3} – y^{3}
 x^{2} + y^{2}
 2xy
 x^{2}y^{2}
Ans: (c)
 The analytic function whose real part is u(x, y) = x^{2} – y^{2} is f (z) =
 z^{2}
 z^{3}
 z^{2}
 ^{2}
Ans: (c)
 The analytic function whose imaginary part is is f (z) =
Ans: (b)
 Cauchy–Riemann equation in polar coordinates
Ans: (c)
 Singularity is a point where f (z) is not
 defined
 having the limit
 continuous
 differentiable
Ans: (d)
 The singularities of cosec z are at z =
 (n + 1/2)π
 (2n – 1)π
 2nπ
 nπ
Ans: (d)
 f (z) = e^{z} has a singularity at
 the origin
 z = πi
 z = nπi
 no point
Ans: (d)
 If f (z) = z/z (z ≠ 0) and f (0) = 1, then f is
 continuous of z = 0 only
 continuous for all z
 discontinuous for all z
 discontinuous at z = 0 only
Ans: (d)
 If u(x, y) = 2x – x^{2} + my^{2} is harmonic in D, then m =
 0
 1
 2
 3
Ans: (b)
 If f (z) = z, then f′(z) exists
 for all z
 nowhere
 at z = 0 only
 at z = 1 only
Ans: (c)
 A function u(x, y) having continuous second partial derivatives and satisfying ∇^{2}u = 0 is called a
 harmonious function
 harmonic function
 holomorphic function
 regular function
Ans: (b)
 The analytic function whose imaginary part is v(x, y) = 2xy is
 z^{2}
 z
 l/z^{2}
 z^{2}
Ans: (d)
 The harmonic conjugate of v(x, y) = 2xy is
 x^{2} – y^{2}
 x – y
 x^{2} + y^{2}
 x^{3} – y^{3}
Ans: (a)
 1/(1 + i) = a + bi ⇒ (a, b) =
 (1/2, –1/2)
 (–1, 1)
 (1,–1)
 (–1/2, 1/2)
Ans: (a)
 (1 + i)/(1 – i) = a + bi ⇒ (a, b) =
 (1,–1)
 (–1, 1)
 (0, 1)
 (1, 0)
Ans: (c)
 Re (e^{z}) =
 e^{x}
 e^{–x}
 e^{x} cos y
 e^{–x} cos y
Ans: (c)
 The complex conjugate of (1 + i)^{2} is
 i
 2/i
 2i
 2 + i
Ans: (b)
 A function u(x, y) having continuous second partial derivatives and satisfying ∇^{2}u = 0 is called a
Ans: (b)
 The analytic function whose imaginary part is v(x, y) = 2xy is
 z^{2}
 z
 l/z^{2}
 z^{2}
Ans: (b)
 The harmonic conjugate of v(x, y) = 2xy is
 x^{2} – y^{2}
 x – y
 x^{2} + y^{2}
 x^{3} – y^{3}
Ans: (a)
 Re (e^{z}) =
 e^{x}
 e^{–x}
 e^{x} cos y
 e^{–x} cos y
Ans: (c)
 If f (z) = z/z (z ≠ 0) and f (0) = 1, then f is
 continuous of z = 0 only
 continuous for all z
 discontinuous for all z
 discontinuous at z = 0 only
Ans: (d)
 If u(x, y) = 2x – x^{2} + my^{2} is harmonic in D, then m =
 0
 1
 2
 3
Ans: (b)
 If f (z) = z, then f′(z) exists
 for all z
 nowhere
 at z = 0 only
 at z = 1 only
Ans: (c)
 The complex conjugate of (1 + i)^{2} is
 i
 2/i
 2i
 2 + i
Ans: (b)
 If f (z) = z(2 – z), then f (1 + i) =
 0
 i
 –i
 2
Ans: (b)
 If f (z) = z then f (3 – 4i) =
 0
 5
 –5
 12
Ans: (b)
 e^{2nπi} =
 0
 –1
 1
 i
Ans: (b)
 If f (z) = u + iv is analytic then
 u_{x} = V_{y}, u_{y} = V_{x}
 u_{x} = V_{y}, u_{y} = –V_{x}
 u_{x} = –V_{y}, u_{y} = V_{x}
 u_{x} = V_{x}, u_{y} = –V_{y}
Ans: (c)
Chapter 4 Complex Integration
 A line integral of any complex function depends
 only on the initial point of the path
 only on the terminal point of the path
 only on the end points of the path
 on the end points as well as the choice of the path
Ans: (d)
 The line integral of a complex function is independent of path if
 the function is analytic in a domain containing the path
 the domain is simplyconnected
 the function is analytic in a simplyconnected domain containing the path
 the function is continuous in a domain containing the path
Ans: (c)
 The integral of a complex function f (z) vanishes over a path C if f (z) is
 analytic and C is any curve in domain D
 any complex function and C is any curve in domain D
 nonanalytic but C is any closed path
 analytic and C is a closed path in a simply connected domain D
Ans: (d)
 A bounded domain D is one which lies
 between two parallel lines
 in some circle about the origin
 outside the unit circle z = 1
 between a pair of intersecting lines
Ans: (b)
 Among the following results, the one that does not follow from Cauchy’s Integral Theorem is
Ans: (d)
 where C lies in . Cauchy’s Theorem is not applicable because
 f (z) ∉ H(D)
 D is not simplyconnected though f (z)∊ H(D)
 f (z) ∊ H(d) but f (z) is not singlevalued
 f (z) ∉ H (nbd of zero)
Ans: (b)
 The value of over a simple closed curve C enclosing πi is
 –2iπ sinh π
 π sinh π
 2 π sinh π
 2 πi sinh π
Ans: (c)
 If C is the circle z = 4, then the value of the integral of over C (counterclockwise) is
 0
 –2πi
 2πi
 1
Ans: (b)
 If over positivelyoriented simple closed curve C about ‘a’, then m =
 –1
 0
 ≠ –1
 1
Ans: (a)
 If C is the st. line segment from 0 to 1 + i, then an upper bound (by MLinequality) for the absolute value of the integral of f (z) = z^{2} over C is
 2
 1
 ∞
Ans: (c)
 If C is a simple closed curve enclosing the origin, then
 πia
 2πia
 2πa
Ans: (b)
 If C is a simple closed curve enclosing the origin, then
 2π
 π
 –2πi
 2πi
Ans: (c)
 where C is a closed path, is
 0
 πi
 2πi
 2π
Ans: (c)
 If C is any simple closed path is true for f (z) =
 sec z
 e^{z}
Ans: (c)
 is true if C is
 z = 2
 z = π
 z = 1
Ans: (d)
 where C : z=1.5, is
Ans: (a)

Ans: (b)

 2i sinh π
 2 sinh π
 2i sin π
Ans: (a)

 1
 2πi
 –i
 0
Ans: (d)

 2πi
 πi
 4πi
Ans: (c)
Hint:
 is true if C is the circle
 z = 1
 z = 2
 z = 4
 none of these
Ans: (a) or (b)
 If C is the straight line from z = 0 to z = i, then
 i/2
 1/2
 i
 –i/2
Ans: (a)
 By MLinequality, if C is the straight line from 0 to 1 + i we have
 2
 4
Ans: (a)

 2 sinh πi + 2πi
 zi sinh πi + π
 2 sinh π + πi
 2i(sinh π + π)
Ans: (d)

 2
 –2
 √2
 π
Ans: (b)
 If C is the unit circle C : z—i = 1 touching the zaxis at the origin, then
 0
 2πi
 πi
 πi/2
Ans: (a)
 Among the following, which one is an entire function?
 e^{z}/z
 z e^{z}
 sec z
 z cosec z
Ans: (b)
 The singularities of (z^{2} +1)/(z^{2} – 1) are at z =
 ±1
 ±i
 1, i
 –1, –i
Ans: (a)
 If where C is the circle z = 2, then f (3) =
 πi
 2πi
 –1
 0
Ans: (d)
 If where C is the circle z = 2, then f (1) =
 2πi
 πi
 4πi
 0
Ans: (c)
 If C is the circle z = π traced in counterclockwise direction
 2σ
 2σi
 σi
 4σi
Ans: (b)
 If C is the unit circle z = 1, then
 0
 1
 2πi
 4πi
Ans: (a)
 If C is the unit circle z = 1, then
 0
 1
 2πi
 4πi
Ans: (c)
 If C is the unit circle z = 1 described in the counterclockwise direction, then
 πi
 2πi
 π/4
 π/8
Ans: (d)
 If C is the circle z = 2 described in the anticlockwise direction, then
 1
 –1
 0
 2
Ans: (b)

 π
 i π/2
 i π
 0
Ans: (c)
 The order of the pole of (e^{z} – 1)/z^{4} is
 3
 1
 2
 4
Ans: (a)
 The residue of f (z) = z^{2}/(z − 1)^{2}(z + 2) at the simple pole z = –2 is
 0
 1
 2/9
 4/9
Ans: (d)

 2πi
 0
 2π
 4πi
Ans: (b)
 If C is the unit circle z = 1, then
 2π
 π
 –π
 –2π
Ans: (a)

 2πi
 πi
 –πi
Ans: (b)

 π
 –1
 0
 1
Ans: (c)
 Along AB: y = 0, 0 ≤ x ≤ 1,
 1/3
 1/2
 2/3
 1
Ans: (a)
 The set {z ∊ C/1 π z π 2} is a
 domain
 closed region
 simplyconnected
 none of these
Ans: (a)

Ans: (a) or (b)

Ans: (a)

 π
 i π/2
 i π
 0
Ans: (b)
 The order of the pole of (e^{z} – 1)/z^{4} is
 3
 1
 2
 4
Ans: (a)
 The residue of f (z) = z^{2}/(z − 1)^{2}(z + 2) at the simple pole z = –2 is
 0
 1
 2/9
 4/9
Ans: (d)

 2πi
 0
 2π
 –2π
Ans: (b)
 If C is the unit circle z = 1, then
 2π
 π
 –π
 –2π
Ans: (b)
 If C is the circle z = π traced in counter clockwise direction
 2π
 2πi
 πi
 4πi
Ans: (b)
 If C is the unit circle z = 1, then
 0
 1
 2πi
 4πi
Ans: (a)
 If C is the unit circle z = 1, then
 0
 1
 2πi
 4πi
Ans: (c)
 If C is the unit circle z = 1 described in the counterclockwise direction, then
 πi
 2πi
 π/4
 π/8
Ans: (d)
 If C is the circle z = 2 described in the anticlockwise direction, then where a =
 1
 –1
 0
 2
Ans: (b)
 If C is the unit circle C : z–i = 1 touching the xaxis at the origin, then
 0
 2πi
 πi
 πi/2
Ans: (a)
 Among the following, which one is an entire function?
 e^{z}/z
 ze^{z}
 sec z
 zcosec z
Ans: (b)
 The singularities of (z^{2} +1)/(z^{2} –1) are at z =
 ±1
 ±i
 1, i
 –1, –i
Ans: (a)
 If where C is the circle z = 2, then f (3) =
 πi
 2πi
 –1
 0
Ans: (d)
 If where C is the circle z = 2, then f (1) =
 2πi
 πi
 4πi
 0
Ans: (c)
 is true if C is the circle
 z = 1
 z = 2
 z = 4
 none of these
Ans: (b)
 If C is the straight line from z = 0 to z = i, then
 i/2
 1/2
 i
 –i/2
Ans: (a)
 By MLinequality, if C is the straight line from 0 to 1 + i we have
 2√2
 2
 √2
 4
Ans: (a)

Ans: (d)

 2
 –2
 √2
 π
Ans: (b)
Chapter 5 Complex Power Series
 The power series represents the function
 e^{(z–i)}
 log(z – i)
Ans: (d)
 The series represents the function
 sin z
 cos z
 tan z
 sinh z
Ans: (b)
 Taylor’s series expansion for f (z)=(z–1)/(z+1) about z = 0 is
 1 + 2z + 2z^{2} + 2z^{3} + …
 –1 + z – z^{2} + z^{3} – …
 –1 + 2z – 2z^{2} + 2z^{3} – …
 1 – z + z^{2} – z^{3} + …
Ans: (b)
 The region of convergence of the power series
 z – 1 π 2
 z – 1 π 1
Ans: (c)
 Laurent’s series expansion of f (z) = z^{2}e^{1/z} at z = 0 is
Ans: (c)
 Laurent’s series expansion of in the region 1 π z π 2 is
Ans: (b)
 (Laurent’s). Give the regions of convergence of the two series respectively:
 z ≤ 1, z ≥ 1
 z π 1, z > 1
 z ≤ 1, z > 1
 z π 1, z ≥ 1
Ans: (b)
 Laurent series of for 0 π z π ∞
Ans: (b)
Zeros and Singularities
 z = nπ (n: integer) are the zeros of the function
 cos z
 sinh z
 sin z
 cosh z
Ans: (c)
 (n: integer) are the zeros of the function
 cos z
 sinh z
 sin z
 sinh z
Ans: (a)
 The singularities of coth z are the zeros of
 tan z
 sinh z
 coth z
 cosh z
Ans: (b)
 The singularities of e^{–z} are the zeros of
 e^{z}
 e^{–z}
 log z
 tan z
Ans: (a)
 The singularity of e^{z}/z^{2}(1 – z)^{3} at z = 0 is a/an
 simple pole
 pole of order two
 essential singularity
 nonisolated singularity
Ans: (b)
 The function has at z = 0 a/an
 essential singularity
 removable singularity
 pole of order two
 a point of continuity
Ans: (b)
 Among the following, the function which has simple pole at z = 0 is f (z) =
 (z + 1)/z(z + 2)
 e^{–z}
 sin z/cos z
 (z^{2} – 1)/z^{2}(1 + z^{2})
Ans: (a)
 Among the following, the function which has a removable singularity at z = 0 is
 (1 – cos z)/z
Ans: (c)
 Among the following, the function which has an essential singularity at z = ∞ is
 e^{z}
Ans: (c)
 Among the following, the function which has a pole of order 3 at z = ∞ is
 sin z
 z + e^{z}
 z^{2} + 2e^{z}
 z^{3}
Ans: (d)
 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.
 essential
 isolated
 removable
 nonessential
Ans: (c)
 The singularity of f (z) = (sin z – cos z)^{–1} is at z =
Ans: (d)
 The simple poles of f (z) = (tan z)/z, which lie inside the circle z = 2, are
Ans: (c)
 The number of singularities of f (z) = e^{–z} is
 one
 two
 infinity
 zero
Ans: (d)
 The simple poles of the function f (z) = (z + 1)/z^{3}(z^{2} + 1) are
 0, –1
 ±i
 –1, ±i
 ±1
Ans: (b)
 The number of simple poles of f (z) = z^{4}/(1 + z^{4}) is
 1
 2
 3
 4
Ans: (d)
 The zeros of are at z =
 nπ (n ∊ z)
 ±nπ (n = 1, 2, 3, …)
 nπ (n = 1, 3, 5, …)
 ±nπ (n = 1, 3, 5, …)
Ans: (b)
 z = 0 is a zero of
 sin z
 cos z
Ans: (c)
 The function f (z) = z^{k} sin z has a zero of second order if k =
 0
 1
 2
 –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
 The function has a pole of order at z = 0
 one
 two
 three
 four
Ans: (b)
Hint:
 The type of singularity that f (z) = (1 –cos z)/z has at z = 0 is a/an ____ singularity.
 essential
 irremovable
 removable
 nonisolated
Ans: (c)
 The zeros of are at z =
 (nπ)^{–1} (n ∊ Z))
 (2n + 1)π (n ∊ Z)
 (nπ)^{–1} (n = 1, 2, 3, …)
 ±(nπ)^{–1} (n = 1, 2, 3, …)
Ans: (d)
 The singularity of is at z =
 –2
 2
 3
 0
Ans: (a)
 The type of singularity at z = 3 for f (z) = (z + 2) sin[(z – 3)^{–1}] is ______ singularity.
 removable
 essential
 pole of order m
 non isolated
Ans: (b)
 The type of singularity at z = 0 for f (z) = (z – sin z)/z^{2} is ____ singularity.
 removable
 essential
 pole of order m
 non isolated
Ans: (b)
 The circle inside which all the zeros of lie is z =
Ans: (d)
 The circle inside which all the zeros of
lie is z =
Ans: (b)
 z = 0 is a removable singularity for the function f (z) =
Ans: (c)
 z = 0 is an essential singularity for the function f (z) =
Ans: (a)
 Among the following, the function which has
infinite number of isolated singularities is
 e^{z}/(z^{2} + 1)
 (sin z)^{–1}
 e^{–z}
Ans: (b)
Chapter 6 Calculus of Residues

 2
 0
 1
 3
Ans: (c)
 If z = a is a simple pole of f (z), then Res [f (z) : a] =
 [(z − a)^{2}f (z)]_{z=a}
Ans: (d)
 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} =
 p′(a)/q(a)
 p(a)/q(a)
 p(a)/q′(a)
 p′(a)/q′(a)
Ans: (c)
 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}=
 g^{(m)}(a)/(m − 1)!
 g^{(m)(a)}/m!
 g^{(m−1)}(a)/(m – 1)!
 g^{(m−1)}(a)
Ans: (c)
 Res {(z + 1)/(z^{2} – 2z) : 0} =
 −1
 1
Ans: (a)
 If f (z) = (1 + e^{z}) / (z cosz + sinz), then Res {f (z) : 0} =
 1
 0
 −1
 2
Ans: (a)
 The residue of cot z at z = 0 is
 0
 1
 −1
 2
Ans: (b)
 If then Res {f(z) : 1} =
 2
 e^{z}
 1
 2e^{2}
Ans: (d)
Hint:
 If , then Res {f (z) : πi} =
 1
Ans: (c)
 If , then a =
 πi
 −πi
Ans: (d)
Hint:
 If then C is a simple closed path such that C
 C_{1} : z = .5
 C_{2} : z = 1.5
 C_{3} : z = 2
 C_{2} U C_{3}
Ans: (d)
 If C is any simple closed path described in counterclockwise direction such that 0 and 1 lie inside C, then
 2πi
 −2πi
 −6πi
 −8πi
Ans: (c)
 If C is any positivelyoriented single closed path such that 0 is inside and 1 is outside, then
 2πi
 −2πi
 −6πi
 −8πi
Ans: (d)
 If C is any positivelyoriented simple closed path such that 1 is inside and 0 is outside, then
 2πi
 −2πi
 −6πi
 −8πi
Ans: (a)
 The value of where C : z = 3/2 is
 tan1
 2 tan1
 2πi tan1
 0
Ans: (c)
Hint:
 Res_{z = 0}(ze^{π/z}) =
 π^{2}
 π^{2}/2
 3π^{2}/2
 0
Ans: (b)
Hint:
 Res_{z = 2i}[ze^{πz}/(z^{2} − 16)] =
Ans: (c)
Hint:
 The value of dz where C is a positivelyoriented simple closed curve enclosing z = ±2i only is
 πi/4
 −πi/4
 −πi/8
 −πi/16
Ans: (b)
 The value of where C is any simple closed path is
 0
 2πi
 4πi
 6πi
Ans: (a)
 Res_{z=−2}[z^{2}/(z − 1)(z + 2)^{2}] =
 1
 0
 8/9
 4/9
Ans: (c)
Hint:
 Res_{z=2}[z^{2}/(z − 1)(z − 2)^{2}] =
 1
 0
 4
 2
Ans: (b)
Hint:
 For evaluating an integral of the type the contour to be used is
 the semi circle s : z = Re^{iθ}(0 ≤ θ ≤ π) in the upper halfplane with the line segment L[−R, R] along the real axis
 the rectangle with vertices at ±a±bi
 the sector z = Re ^{iθ}(0 ≤ θ ≤ α)
 the unit circle z = 1
Ans: (d)
 For evaluating an integral of the type the contour to be used is
 the semicircle s : z = Re ^{iθ}(0 ≤ θ ≤ π) in the upper halfplane with the line segment L[R, R] along the real axis
 the rectangle with vertices at ±a ± bi
 the sector s = z = Re^{iθ}(0 ≤ θ ≤ π)
 the unit circle z = 1
Ans: (a)
 If C is the circle z = π described in counterclockwise direction, then =
 πi
 2πi
 0
 −1
Ans: (b)
 If C is the circle z = , then
 πi
 2πi
 0
 −1
Ans: (c)
 If C is the circle z = 1, then
 πi
 2πi
 0
 −1
Ans: (b)
 If C is the circle z = 3, then
 πi(e^{4} − e^{2})
 2πi(e^{4} − e^{2})
 0
 −1
Ans: (b)
Hint:
 Let f(α) If z = 2,α = 1, then F(1) =
 4πi
 2πi
 0
 6πi
Ans: (a)
 If z = , α = 1, then F(1) in Qn 28 is
 4πi
 2πi
 0
 6πi
Ans: (c)
 For any α lying inside C (counterclockwise), we have F(α) =
 −2πi(α^{2} + 1)
 2πiα, 2πi
 0
 πi(α^{2} + 1)
Ans: (b)
 F′(α) and F"(α) in Qn 30 are
 4πiα, 4πi
 2πiα, 2πi
 0
 πi(α^{2} + 1)
Ans: (a)
Chapter 7 Argument Principle and Rouche’s Theorem
 A function f (z) whose only singularities are poles is called ______ function
 isomorphic
 meromorphic
 endomorphic
 epimorphic
Ans: (b)
 Among the following, the function which is meromorphic is
 e^{z}
 sin z
 tan z
 cos z
Ans: (c)
 Let f, g ∈ H(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.
 g (z) π f (z)
 g (z) = f (z)
 g (z) > f (z)
 f (z)g(z) > 1
Ans: (a)
 Let f, g ∈ H(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.
 g(z)
 g (z) + f (z)
 g (z) – f (z)
 g (z)f (z)
Ans: (b)
 The name of the theorem stated in Qn 4 is ________ theorem.
 Rouche’s
 Argument
 Liouville
 Fundamental theorem of Algebra
Ans: (a)
 The number of roots of z^{6} + 16z + 1 = 0 that lie inside z = 2 is
 2
 4
 5
 6
Ans: (d)
Hint: f(z) = z^{6} (has 6 zeros) g(z) = 16z + 1 z^{6} + 16z 1 has the same no. of zeros as f, i.e., 6.
 The number of zeros of z^{6} + z^{5} + 1 inside C : z = 2 is
Ans: (c)
Hint: f(z) = z^{6} (six zeros) g(z) = z^{5} + 1 z^{6} + z^{5} + 1 has the same no. of zeros as f, i.e., 6.
 The number of zeros of z^{6} + z^{5} + 1 inside C : z = 2 is
 4
 3
 6
 0
Ans: (b)
Hint: f(z) = 2z^{5} + 1, g(z) = : z^{6} + 5z^{5} + 1 has the same no. of zeros as f, i.e., 5
 The number of zeros of z^{n} + z^{5} + 1 inside C : z = 1 is
 0
 1
 2
 n
Ans: (d)
Hint: f(z) = z^{n} (it as n zeros); g(z) = z + 1 has the same no. of zeros as f(z), i.e., n.
 The number of zeros of z^{n} + z^{5} + 1 inside C : z = 1 is
 0
 1
 2
 n
Ans: (b)
Hint: f(z) = z + 1 (are zero) g(z) = z^{n} has the same no. of zeros as f(z), i.e., 1.
 The number of zeros of z^{n} +z+1 in the annulus 1 ≤ z ≥ 2 is
 2
 n
 3
 (n−1)
Ans: (d)
 The number of zeros of 3z^{n} − e that lie inside z = 1 is
 1
 2
 n
 n−1
Ans: (c)
 All the roots of πz^{2} − e^{z} + 0.2 lie inside the circle z =
 0.1
 0.25
 0.5
 1
Ans: (d)
Hint:
 z^{7}−5z^{3} + 12z^{2} has ________ zeros in the annulus 1 ≤ z ≤ 2.
 7
 6
 5
 0
Ans: (c)
Hint:
 The number of roots of z^{7} − 5z^{3} = 0 that lie in the punctured disc 0 π z π 3/2 is
 7
 4
 3
 0
Ans: (b)
Hint:
Chapter 8 Conformal Mapping
 A mapping that preserves angles between oriented curves both in magnitude and in sense is called a/an ________ mapping.
 informal
 isogonal
 conformal
 formal
Ans: (c)
 The mapping defined by an analytic function f (z) is conformal at all points z except at points where ______
 f’(z) = 0
 f’(z) ≠ 0
 f’(z) > 0
 f’(z) π 0
Ans: (a)
 Under the mapping w = z^{2}, a circle z = c is transformed to a/an
 ellipse a^{2}u^{2} + b^{2}v^{2} = 1
 circle w = c^{2}
 vertical line
 horizontal line
Ans: (b)
 The mapping w = z^{2} transforms a vertical line to a/an
 horizontal line
 circle
 ellipse
 parabola
Ans: (d)
 Under the mapping w = f (z) if the angle θ = π/n is transformed to an angle π in the wplane, then f (z)
 z^{n}
 z^{2n}
 z^{1/n}
 z/n
Ans: (a)
 Under the transformation w = iz, the st. line y = x in the zplane is rotated through an angle _______in the wplane.
Ans: (c)
 The mapping w = preserves angles in
 size but not in sense
 sense but not in size
 size as well as sense
 neither size nor sense
Ans: (a)
 The mapping w = z^{2} is not conformal at
 any point
 any point except z = 0
 the origin
 any point along the real axis
Ans: (b)
 e^{z} maps
 vertical lines into circles
 circles into vertical lines
 circles into circles
 straight lines into straight lines
Ans: (a)
 e^{z} maps
 circles into circles
 circles into horizontal lines
 horizontal lines into rays through O
 rays through O into circles
Ans: (c)
 The two points where the mapping is not conformal are z =
 0, 1
 ±∞
 0, 0
 ±1
Ans: (d)
 The images of the vertical lines under the mapping w = sin z are
 hyperbolas
 ellipes
 circles
 parabolas
Ans: (a)
 The images of the horizontal lines under the mapping w = sin z are
 hyperbolas
 ellipes
 circles
 parabolas
Ans: (b)
 The condition for the map w = (az + b)/(cz + d) to be conformal for all z is that
Ans: (c)
 The transformation w = (az + b)/(cz + d) maps the unit circle w = 1 into a st. line in the zplane if
 a = c
 a = d
 b = c
 b = d
Ans: (a)
 The fixed points of the transformation w = z^{2} are
 0, 1
 0, −1
 −1, 1
 −i, i
Ans: (a)
 The invariant points of the mapping w = are
 1, −1
 0, −1
 0, 1
 −1, −1
Ans: (c)
 The fixed points of w = are
 ±1
 ±i
 0, −1
 0, 1
Ans: (b)
 The mapping w = transforms circles of constant radius into
 confocal ellipses
 hyperbolas
 circles
 parabolas
Ans: (a)
 Among the following, the one that is not a critical point of the mapping w = cos z is
 π
 2π
 3π
 π/2
Ans: (d)
 The linear fractional transformation that maps the points ∞, 1, 0, respectively, into 0, 1, ∞ is w =
 z
 z^{−1}
 iz
 −iz
Ans: (b)
 The mobius transformation that maps the points 0, i, ∞, respectively, into 0, 1, ∞ is w =
 z^{−1}
 −z
 −iz
 iz
Ans: (c)
 The images of the points z_{1} = ∞, z_{2} = i, z_{3} = 0 under the transformation w = −1 / z are
 0, i, ∞
 0, − i, ∞
 0, −i, −∞
 0, −i, 1
Ans: (c)
 The fixed points of the transformation w = are
 1, −1, −i
 i, 1, −i
 1, −1, i
 i, −1, −i
Ans: (d)
 Under the transformation w = , the point is mapped into the point
Ans: (b)
 The image of the line x = 1 under the transformation w = sin z is a/an
 circle
 ellipse
 hyperbola
 parabola
Ans: (c)
 The image of the line y = c under the transformation w = sin z is a/an
 circle
 ellipse
 hyperbola
 parabola
Ans: (b)
 The mapping under which a vertical line z = c transforms into a circle is w =
 sin z
 cosh z
 z^{2}
 e^{z}
Ans: (d)
 z = 0 is a critical point of the transformation w =
 sin z
 z^{n}(n > 1)
 e^{z}
Ans: (b)
 The mapping under which vertical lines are transformed into a circle is w =
 z^{2}
 e^{z}
 sin z
Ans: (c) z
 Under the mapping w = z^{2}, a st. line through the origin is transformed into a/an
 circle
 ellipse
 parabola
 st. line through O
Ans: (d)
 Under the mapping w = z^{2}, a circle of radius 3 is transformed into a circle with radius
 √3
 3
 9
 6
Ans: (c)
 Under the mapping, the circle of radius ________ is transformed into a circle with radius .
 1
Ans: (c)
 The image of the circles  w − 1 = k under the mapping w = z^{2} is given by
 z − 1z + 1= k
 z − 1= k
 z + 1= k
 z^{2} = k
Ans: (a)
 The fixed points of the transformation w = z/(z + 1) are
 0, −1
 0, 1
 1, −1
 2, 1
Ans: (a)
 The sector 0 ≤ 0 ≤ is mapped onto the upper half−plane by w = z^{n} if n =
 3
 6
 3
 4
Ans: (b)
 The number of fixed points of the mapping w = is
 1
 2
 3
 4
Ans: (b)
 The equation of the line whose image under the map w = e^{z} is the circle with radius with radius ‘e’ is
 x = 0
 y = 1
 y = 0
 x = 1
Ans: (d)
 The ______ mapping has every point of it as a fixed point.
 rotation
 inverse
 identity
 traslation
Ans: (c)
 w = has ________ fixed points.
 no
 infinitely many
 finitely many
 some
Ans: (b)
 The translation mapping w = z + c (c ≠ 0) has _________ fixed points.
 no
 infinitely many
 finitely many
 some
Ans: (a)
 The rotation mapping w = cz has ________ fixed point(s).
 no
 infinitely many
 finitely many
 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
 _______.
Ans:
 _______.
Ans: 3/2
 _______.
Ans: 2
 _______.
Ans:
 _______.
Ans:
 _______.
Ans: π cosec pπ
 B(1/3, 2/3) = _______.
Ans:
 B(1/2, 1/2) = _______.
Ans: π
 _______.
Ans: –2
 _______.
Ans: 1/2
 B(1, 1/2) = _______.
Ans: 2
 Γ(3/2)Γ(1/2) = _______.
Ans: π/2
 B(p + 1, q) + B(p, q + 1) = _______.
Ans: (p, q)
 B(p + 1, q)/B(p, q) = _______.
Ans: p/p + q
 Γ(1/2) = _______.
Ans:
 Γ(3/2) = _______.
Ans:
 Γ(–1/2) = _______.
Ans:
 Γ(x) = x ⇒ x = _______.
Ans: 1
 Γ(–3/4) = _______ Γ(1/4).
Ans: –4/3
 Γ(1/4)Γ(3/4) = _______.
Ans:
 Γ(1/3)Γ(2/3) = _______.
Ans:
 _______.
Ans: 1/3
 _______.
Ans:
 _______.
Ans: 2^{2p–1}
 Γ(p)Γ(1 – p) = π ⇒ p = _______.
Ans: 1/2
 _______.
Ans: Γ(n)/a^{n}
 _______.
Ans: 2
 _______.
Ans: Γ(1/p + 1)
 If tan α = 1 (0 π α π π/2) and Γ^{2}(x) = α then x = _______.
Ans: 3/2
 _______.
Ans:
 _______.
Ans: π
 (m + n) B (m + 1, n) = _______.
Ans: mB(m, n)
 _______.
Ans: π
 _______.
Ans:
 _______.
Ans: (1/2, 1/2)
 _______.
Ans: B(2, 3)
 _______.
Ans:
 _______.
Ans: B(5, 6)
B. Legendre Functions
 [P1(–x)]^{1} = ________.
Ans: –1
 ________.
Ans: 2/(2n + 1)
 Rodringue’s Formula is P_{n}(x) = CD^{n}[(x^{2} – 1)^{n}] with C = ________.
Ans: 1/(2^{n}n!)
 The generating function for Pn(x) is ________.
Ans: (1 – 2xt + t^{2})^{–1/2}
 The sum of the series 1 + t + t^{2} + t^{3} + … in terms of Legendre function is ________.
Ans: P_{n} (1)
 P_{0}(x) = ________.
Ans: 1
 ________ (m ≠ n).
Ans: 0
 P_{1}^{1}(x) = ________.
Ans: P_{0} (x)
 P_{n}(1) = ________.
Ans: 1
 P_{n}(–x) = ________ P_{n}(x).
Ans: (– 1)^{n}
 P_{n}(– 1) = _________
Ans: (– 1)^{n}
 ________.
Ans: 2/7
 ________.
Ans: 0
 If x^{2} + x = aP_{0} + bP_{1} + cP_{2}, then (a, b, c) = ________.
Ans:
 P_{2n}(0) = _______
Ans: (–1)^{n}(2n)!/2^{2n}(n!)^{2}
 ________.
Ans:
 Legendre’s differential equation is ________.
Ans: [(1 – x^{2})y′]′ + x(x + 1)y = 0
 (n + 1)P_{n+1}(x) + nP_{n–1}(x) = ________.
Ans: (2n + 1)xP_{n}(x)
 P′_{n+1} (x) – P′_{n – 1}(x) = ________.
Ans: (2n + 1)P_{n}(x)
 ________.
Ans: 2n(n + 1)/(2n + 1)
C. Bessel Functions
 Physical and Engineering Problems involving vibrations or heat conduction in cylindrical regions give rise to _________ equation.
Ans: Bessel’s differential
 Bessel’s differential equation of order n is ________.
Ans:
 Solutions of Bessel’s differential equation are ________called of order n.
Ans: Bessel’s functions
 The series expansion of J_{n} (x) = ________.
Ans:
 J_{n}(x) is called _________.
Ans: Bessel function of the first kind of order n
 When n is an integer a second linearly independent solution of Bessel’s differential equation is ________.
Ans:
 _________ is called the Bessel function of the second kind of order n or Neumann function.
Ans: y_{m}(x)
 J_{1/2}(x) = ________.
Ans:
 J_{–1/2}(x) = ________.
Ans:
 ________.
Ans: J_{n+1} (x)
 The generating function for J_{n} (x) is ________.
Ans:
 ________.
Ans: J_{n}(x)
 ________.
Ans: J_{2m} (x)
 J^{2}_{0} + 2(J^{2}_{1} + J^{2}_{2} + …) = ________.
Ans: 1
Chapter 2 Functions of a Complex Variable
 f (z) = is ________.
Ans: not analytic for any z
 The Cauchy−Riemann equation is only ________ but not ________ for analytic of f (z)
Ans: necessary, sufficient
 f (z) = z^{2} is ________.
Ans: continuous for all z but derivable at z = 0 only
 is ________.
Ans: analytic for all finite z ≠ 1
 The area of the triangle formed by the points 1 + i, i – 1, 2i (in sq. units) is ________.
Ans: 1
 The triangle formed by joining the points (±1 +) and the origin is ________.
Ans: equilateral
 The discontinuity of at z = −i may be removed by defining f (–i) = ________.
Ans: i
 An analytic function of constant absolute value is ________.
Ans: constant
 A function u(x, y) having continuous second partial derivatives and satisfying ∇^{2}u = 0 is called a ________ function.
Ans: harmonic
 The analytic function whose imaginary part is v(x, y) = 2xy is ________.
Ans: z^{2}
 The harmonic conjugate of v(x, y) = 2xy is ________.
Ans: x^{2} – y^{2}
 If f(z) = z/z (z ≠ 0) and f(0) = 0, then f is ________
Ans: discontinuous at z = 0 only
 If u(x, y) = 2x – x^{2}+my^{2} is harmonic in D, then m = ________.
Ans: 1
 If f (z) = z , then f ’(z) exists ________.
Ans: z = 0 only
 If f (z) = u + iv is analytic, then ________.
Ans: U_{x} = V_{y}, U_{y} = – V_{x}
 f (z) = (1 + z)/(1 – z) is differentiable at ________.
Ans: all z except z = 1
 f (z) = Re(z) is differentiable at ________.
Ans: no z
 The curves u(x, y) = e^{x} cos y = c′ are orthogonal to the curves v(x, y) = c_{2} where v(x, y) = ________.
Ans: e^{x} sin y
Chapter 3 Elementary Functions
 sin(x + iy) = ________.
Ans: (sin x cosh y + i cos x sinh y)
 cos(x + iy) = ________.
Ans: (cos x cosh y – i sin x sinh y)
 Re[tan(x + iy)] = ________.
Ans: sin2x/(cos2x + cosh y)
  cos(x + iy)^{2} = ________.
Ans: (cos^{2} x + sin^{2} y)
 Sin z is periodic with period ________.
Ans: 2
 Tan z is periodic with period ________.
Ans: π
 e^{z} is periodic with period ________.
Ans: 2πi
 ________.
Ans:
 ________.
Ans:
 The solutions of cos z = 0 are z = ________.
Ans: (2n + 1)
 When expressed in terms of natural logarithm, sin^{–1} z = ________.
Ans:
 When expressed in terms of natural logarithm, cos^{–1} z = ________.
Ans:
 All the roots of cos z = 2 are given by z = ________.
Ans:
 All the roots of sinh z = i are given by z = ________.
Ans:
 When expressed in terms of logarithm, tan^{–1} z = ________.
 When expressed in terms of logarithm, sec^{–1} z = ________.
Ans:
Chapter 4 Complex Integration
 ________.
Ans: 1 – i
 where C is the upper half of the unit circle taken in the clockwise direction, is ________.
Ans: —πi
 If C is the circle z – a = r, then ________.
Ans: 2πi
 If C is the circle z – a = r and n ≠ –1, then ________.
Ans: 0
 If C is the straight path y = x from z = 0 to 1 + i, then ________.
Ans:
 ________.
Ans:
 ________.
Ans:
 ________.
Ans:
 If C is the arc of the circle z = 2 from θ = 0 to θ = π/3, then ________.
Ans:
 If C is the left half of the unit circle in the clockwise direction, then ________.
Ans: 2i
 By the MLinequality, an upper bound for is ________.
Ans:
 By the Cauchy’s Integral Theorem, ________ (C : z = 1).
Ans: 0
 By the Cauchy’s Integral Theorem, ________ (C : z = 1).
Ans: 0
 By the Extended Cauchy Integral Theorem for multiplyconnected domains, (C : z = 2) = ________.
Ans: 4πi
 By the Cauchy’s Integral Theorem, (C : z – i = 2) = ________ (C : z = 1).
Ans:
 If C is the circle z = 2, ________.
Ans:
Chapter 5 Complex Power Series
 For z > 1, (1 + z)^{–1} = ________.
Ans:
 Taylor’s series about z = 0 of is ________.
Ans:
 The region of convergence of the series in Qn. 2 is ________.
Ans: R : z π 1
 Laurent’s series expansion about z = 1 of is ________.
Ans:
 The region of convergence of the series in Qn. 4 is ________.
Ans: R : 0 π z – 1 π 2
 Taylor’s series for about z = 0 is ________.
Ans:
 The region of convergence of the series in Qn. 6 is ________.
Ans: R : 0 π z π 1
 Laurent’s series expansion for z > 1 of f (z) = is ________.
Ans:
 Laurent’s series expansion for z – 1 > 1 of is ________.
Ans:
 Laurent’s series expansion for z > 2 of f (Z) = is ________.
Ans:
Chapter 6 Calculus of Residues
 The poles of are z = ________.
Ans: ±i
 The Residues of is ________.
Ans:
 The Residues of is ________.
Ans:
 If C is the circle z = 2, then by the Residue Theorem ________.
Ans: 2πi
 The pole of ________.
Ans: 1
 The order of the pole in Qn. 5 is ________.
Ans: 3
 The Residue at z = 1 of f (z) = is ________.
Ans:
 By the Residue Theorem, where C : z = 2 is ________.
Ans: 3πie
 The poles of are z = ________.
Ans: 1, 2
 The Residue of f(z) at z = 1 in Qn. 9 is________ .
Ans: 1
 If C : z = 3/2, then ________.
Ans: 2πi
 If C: z = 5, then ________.
Ans: 2πi cos 3
 ________.
Ans:
 ________.
Ans: π/12
 ________.
Ans: π/6
Chapter 7 Argument Principle And Rouche’S Theorem
 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
 A point z = a where f(z) is not differentiable is called a/an ________ of f(z).
Ans: singularity
 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
 If z = a is an isolated singularity for f (z) such that (z – a)^{m}f (z) = ϕ(D) ∈ H(D) for some positive integer m and ϕ(a) ≠ 0 is called a ________.
Ans: pole of order m
 If m = 1 in Qn. 4, then it is called a ________.
Ans: simple pole
 A complex function in a domain D whose singularities are only poles is called ________function.
Ans: meromorphic
 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:
 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]
 Rouche’s theorem states that if f, g ∈ H(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)
 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
 A mapping that preserves the angles between oriented curves in magnitude and sense is called a ________.
Ans: conformal mapping
 The points which are mapped onto themselves under a conformal mapping are called ________ points.
Ans: fixed invariant
 The fixed points of = ________.
Ans: [0, 1]
 The fixed points of z = ________.
Ans: ±i
 The fixed points of _______.
Ans: [3, 3]
 The fixed points of z = ________.
Ans: – 1 ± 2i
 The critical points of the mapping w = z^{2} is z = ________.
Ans: 0
 The critical points of the mapping w = z + ________.
Ans: ±1
 The invariant points of the transformation w = are Z = ________.
Ans: ±i
 The invariant points of the transformation w = are Z = ________.
Ans: 3i ± 2
 The transformation w = sin z maps st. lines x = c into the family of ________.
Ans: confocal hyperbolas
 For the transformation w = f(z), the point where f’ (z) = 0 is called a ________ point.
Ans: critical
 The transformation w = cosh z maps horizontal lines y = c into the family of ________.
Ans: confocal hyperbolas
 The transformation w = cosh z is not conformal at points where z = ________.
Ans: nπi
 The image of z – 1 = 1 under the mapping ________.
Ans:
 Under the bilinear transformation maps circles into ________ or ________.
Ans: circles, st. lines
 Under the transformation w = z^{2}, the image of the line z = 4 is a ________ whose equation is ________.
Ans: parabola, v^{2} = –64(u –16)
 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. 2^{2p–1} Γ (p) Γ(p + 1/2) 
4. 

5. Γ(2p) 
e. Γ(p)Γ(q)/Γ(p + q) 

f. π sin pπ 
Ans: 1. c 2. a 3. d 4. b 5. c
B. Legendre Functions
A (Function) 
B (Value) 
1. P_{2n+1}(0) 
a. (–1)^{n}(2n)!/2^{2n}(n!)^{2} 
2. P2n(0) 
b. (–1)^{n} 

c. 
4. 
d. 
5. P_{m}(x)P_{n}(x)dx 


f. 0 
Ans: 1. f 2. a 3. b 4. e 5. c
C. Bessel Functions
A 
B 
1. 
a. 

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 

a. 

b. 
3. u(x, y) = 2 log(x^{2} + y^{2}) + k 
c. u_{x} = v_{y}; u_{x} = –v_{y} 
4. u(r, θ) = r^{2} cos 2θ = c 
d. u is a harmonic function 
5. 
e. v(r, θ) = r^{2} 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) 

a. 

b. 
3. (1 + i)^{i} 
c. 

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. 

2. 

3. 

4. 

5. 


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. 

2. 

3. 

4. f(z) = e^{z} 
d. Pole of higher order 
5. 


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. 

3. 

4. Res _{z= ∏/2} tan z 

5. 


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. z^{5} + 15z + 1 = 0 
a. 1 root in each quadrant 
2. e^{z} = az^{4} (a > e) 
b. 1 root in the first quadrant 

c. 
4. z^{4} + 4(1 + i) + 1 = 0 
d. 4 roots lie in the first quadrant 
5. z^{4} + z^{3} + 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. 

5. 


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

T

F

T

F
 Γ(p)Γ(1 + p) = π cosec pπ
F

T
B. Legendre Functions

T
 Rodrigues’ Formula is
T

F
 if n is odd
F
 P_{n}(1) = 1
T
 P_{n}(–x) = (–1)^{n}P_{n}(x)
T
C. Bessel Functions
 The differential equation satisfied by J_{0}(x) is xy″ + y′ + xy = 0
T

F

T

T
 J′_{0}(x) = J_{1}(x)
F
 cos x = J_{0} – 2J_{2} + 2J_{4}
T
Chapter 2 Functions of a Complex Variable
 f (z) = /z is not continuous at z = 0
T
 If a complex function is derivable at a point then it is continuous there.
T
 If a function f (z) = u + iv satisfies Cauchy–Riemann equations, then f (z) is analytic
F
 Laplace equation in polar coordinates
T
 f (x +iy) = 2xy +i(x^{2} – y^{2}) is analytic
F
 is analytic
F
Chapter 3 Elementary Functions
 sin(iz) = i sinh z
T
 (coth z)′ = cosech^{2}z
F
 Principal value of i^{i} is e^{π/2}
F
 A solution of e^{z} = 1 + 2i is z =
T
 The principal value of log(1 + i) + log(1 – i) is log 2
T

F
Chapter 4 Complex Integration
 If C is the line segment from z = 0 to z = i,
F

T
 If C is the unit circle z = 1 then
F

T
 is independent of path c of integration
F
 If C is z = 3 then 8πi
T
Chapter 5 Complex Power Series
 A series of the form is called a power series
T
 If a series ∑ u_{n} (x) converges for every z in C : z – a = R, then C is called the circle of convergence
T
 Taylor’s series expansion of a function f (z) consists of both positive and negative powers of (z – a)
F
 If is expanded in the annular region 1 π z π 2 the series we obtain is Taylor’s series
F
 Taylor’s series is a particular case of Laurent’s series
T
 If f (z) has a Taylor’s series expansion valid in a region D, then f (z) ∈ H (D)
T
Chapter 6 Calculus of Residues
 By Cauchy’s Residue Theorem, real integrals can be evaluated.
T
 If then b_{1} is called the residue of f (z) at z = a
T

F

F

T
 If C is z = 2, then
T
Chapter 7 Argument Principle and Rouche’S Theorem
 Liouville’s Theorem states that if f (z) ∈ H(D) and f (z) is bounded in D then f (z) is constant in D
T
 Every polynomial of degree n has n roots
T
 If f (z) ∈ H (D) then min f (z) or max f (z) can occur at any point of D
F
 z^{5} + 15z + 1 = 0 has all its 5 roots in the unit circle z = 1
F
 e^{z} = z^{n} has n roots inside the unit circle
F
 One root of z^{4} + z^{3} + 1 = 0 lies in the first quadrant
T
Chapter 8 Conformal Mapping
 The fixed points of the mapping is z = i
T
 The mapping w = z^{1/n} maps sectors into half planes
T
 The image of the circle z + 1 = 1 under the mapping is the circle
F
 A bilinear transformation preserves the crossratio of four points
T
 The critical points of w = cos z are at z = nπ
F
 If the mapping w = f (z) is conformal in D, then f (z) ∈ H (D)
T