MA 41006/ MA51108: Complex Analysis/ Advanced
Complex Analysis
Assignment-01 (Spring 2023)
1. Find the following limits (if exists)
(Re(z) − Im(z))2
(a) lim
z→0
|z|2
[
]
1
2
(b) lim
1 + iy
z→0 1 − e x
( )2
z
(c) lim
z→0
z̄
z
(d) lim
z→0 |z|
2. Test the continuity of the following functions:
{
{ 2
{
Im(z)
Re(z 2 )
z
if
z
=
̸
0
if
z
=
̸
0
|z|
|z|
|z|2
(a) f (z) =
(b) f (z) =
(c) f (z) =
0
if z = 0
0 if z = 0
0
if z ̸= 0
if z = 0
3. Examine the continuity of f (z) at z = 0, where f (z) is
{ 2
xy
if z ̸= 0
x2 +y 4
f (z) =
0
if z = 0
4 At which points the following function is C-differentiable?
{ z2
if z ̸= 0,
z
f (z) =
0 if z = 0.
5. Show that
{
f (z) =
x3 (1+i)−y 3 (1−i)
x2 +y 2
0
if z ̸= 0
if z = 0
satisfies Cauchy Reimann equations at z = 0 but f ′ (0) does not exist.
6. Show that for the function,
{ 2
xy (x+iy)
if z ̸= 0
x2 +y 4
f (z) =
0
if z = 0
f ′ (0) does not exist but it satisfies Cauchy Reimann equations at (0, 0).
7. Using Cauchy Reimann equations show that
(a) f (z) = |z|2 is not analytic at any point.
(b) f (z) = z̄ is not analytic at any point.
1
(c) f (z) = , z ̸= 0 is analytic at all points except at the point z = 0.
z
8. Show that the function Logz is analytic for all z except the point {z : Re z ≤
0, Im z = 0}.
1
9. Let f (z) = u + iv be analytic in a domain D. Prove that f is constant in D
if any one of the followings hold.
(a) f ′ (z) vanishes in D.
(b) Re f (z) = u = constant.
(c) Im f (z) = v = constant.
(d) |f (z)| = constant (non zero).
10. Show that the function u = cos x cosh y is harmonic. Find its harmonic
conjugate.
11. Show that following functions are harmonic:
(a) u(x, y) = 2x + y 3 − 3x2 y
(b) v(x, y) = ex sin y
and find their harmonic conjugates and the corresponding analytic functions
f (z).
12. u(r, θ) = r2 cos 2θ is harmonic. Find its conjugate harmonic function and
the corresponding analytic function f (z).
13. If f (z) is analytic function of z, then prove that
∂2
∂2
∂2
(a)
+
=
4
.
∂x2 ∂y 2 ) ∂z∂ z̄
(
∂2
∂2
(b)
+
|f (z)|2 = 4|f ′ (z)|2 .
∂x2 ∂y 2
14. Find the values of constants a, b, c and d such that the function f (z) =
(x2 + axy + by 2 ) + i(cx2 + dxy + y 2 ) is analytic.
15. Suppose f (z) = u + iv is analytic at z0 ̸= 0. Show that
(
)
i ∂u
∂v
′
f (z0 ) = −
+i
z0 ∂θ
∂θ
at z = z0 , where (r, θ) are the polar coordinates.
2
MA 41006/ MA51108: Complex Analysis/ Advanced
Complex Analysis
Assignment-02 (Spring 2023)
1. Discuss the convergence of the complex sequence {zn }, where zn is as given
in the following forms.
(
)
(
)
1
1
n−i
(i) zn = 1 −
+i 2+ 2
(ii) zn =
n
n
n+i
n
2
n
(iv) zn = 2 + i2n
(iii) zn =
+i n
n!
2
nπ
nπ
(v) zn = in cos nπ
(vi) zn = sinh
+ i cosh
4
4
2. Test the convergence of the following series:
∞
∞
∑
∑
π
1 + in
(i)
(ii)
ein 4
3
n
n=1
n=1
)n
∞
∞ (
∑
∑
1
3n − 2
(iii)
(iv)
(3 − 4i)n
n
(1 + i)
np + 1
n=1
n=1
(v)
∞
∑
n=1
zn
n(n + 2)
(vi)
∞
∑
n!z n
n=1
3. Find the radius of convergence of each of the following power series:
( √
)
∞
∞
∑
∑
n 2+i
zn
n
(i)
z
(ii)
1 + 2in
2n + 1
n=0
n=0
(iii)
∞
∑
√
2
∞
∑
(n2 )! n
(iv)
z
(2n)!
n=1
n n
z
n=0
4 Determine the exact region of convergence of each of the following power
series:
)n
∞ (
∞
∑
∑
iz − 1
(z + 2)n−1
(i)
(ii)
2+i
(n + 1)3 4n
n=0
n=1
(iii)
∞
∑
(−1)n z 2n−1
n=1
(iv)
(2n − 1)!
∞
∑
n=2
zn
n(log n)2
∑∞
n
5. If R1∑and R2 are the radii of convergence of the power series
n=0 an z
∞
n
and n=0 bn z∑ respectively, then show that the radius of convergence of the
n
power series ∞
n=0 an bn z is R1 R2 .
1
MA 41006/ MA51108: Complex Analysis/ Advanced
Complex Analysis
Assignment-03 (Spring 2023)
∫
1. Evaluate C |dz|.
∫
2. Find the value of the integral, C z dz from z = 0 to z = 4 + 2i along the
curve C given by (i) z = t2 + it and (ii) the line from z = 0 to z = 2i and
then, the line from z = 2i to z = 4 + 2i.
3. Evaluate the
∫ followingn integrals
(i) I1 = C (z − z0 ) dz, n = 0, ±1, ±2, · · · , where C : |z − z0 | = r is
traversed
in the counterclockwise direction.
∫
(ii) I2 = C (x + y 2 − ixy) dz where
{
t − 2i,
if 1 ≤ t ≤ 2
C : z = z(t) =
2 − i(4 − t), if 2 ≤ t ≤ 3.
4. Prove that the value of the integral of z1 along a semicircular arc |z| = 1 from
−1 to +1 is −πi or πi according ∫as the arc lies ∫above or below the real axis.
5. Obtain the value of the integral C |z|2 dz and C z 2 dz along the curves
(i) C : C1 : z1 (t) = t + it (0 ≤ t ≤ 1) and
(ii) C : C2 : z2 (t)∫ = t2 + it (0
∫ ≤ t ≤ 1). ∫
∫
Also verify that C1 |z|2 dz ̸= C2 |z|2 dz and C1 z 2 dz = C2 z 2 dz.
∫
6. Show that the integral C e−2z dz, where C is the path joining the points
z = 1 + 2πi and z = 3 + 4πi is independent of the path of integration.
Evaluate the integral by taking a suitable path.
∫ 5+3i
7. Using the definition of the integral of f (z) on a given path, evaluate −2+i z 3 dz
∫ 2
and C (z +3z +2) dz where C is the arc of the cycloid x = a(θ +sin θ), y =
a(1 − cos θ) between the points
0) and (πa, 2a).
∫ z (0,
dz
8. Evaluate the integrals I = C 1+z2 , where C is the upper semi circle of the
circle z + 21 = 12 traversed in the anti clockwise direction.
9. By using the Fundamental theorem of integral calculus for complex functions
evaluate
∫ the following integrals.
(i) C (z + a)ebz dz (b ̸= 0), where C is the parabolic arc x2 = y from (0, 0)
to (1, 1).
∫ 1+2i z
ze dz
(ii) 0
(iii)
∫i
−i
5
z 4 eπz dz
∫ 1 −1 z
(iv) 0 tan
dz .
1+z 2
10. Can the Cauchy’s integral theorem be applied for evaluating the following
integrals
evaluate these integrals.
∫ ? Hence,
2
(i) ∫C esin z dz; C : |z| = 1.
(ii) C tan z dz; C : |z| = 1.
1
∫
(iii) C z3dz−1 where C : a triangle with vertices at 0, ± 14 + 2i .
∫ 2
dz; C : |z| = 3.
(iv) C z +5z+6
∫ ezz−2
(v) C z2 +9 dz; C : |z| = 2.
∫
(vi) C z1 dz; C : |z| = 1.
11. Verify Cauchy’s theorem for the functions 4 sin 2z, 3z − 2 and z 2 + 3z − 1 if
C is the square having corners ±1 ± i.
12. Verify Cauchy’s theorem for the function z 3 − iz 2 − 5z + 2i if C is the circle
|z − 1| = 2.
13. Using the Cauchy’s integral theorem and its extension evaluate the following
integrals.
∫
dz
(i) C z(z+2)
, where C is any rectangle containing the points z = 0 and
z∫= −2 inside it.
(ii) C z7z+12
dz, C : |z − 2| = 2.
∫ 2 +z−2
dz,
C : |z| = 1.
(iii) C z3z+5
∫ 2 +2z dz
(iv) C (z−1)(z−2)(z−3) , C : |z| = 4.
∫
(v) C 2z−1
dz, C : |z| = 2.
z 2 −z
14. Using∫Cauchy’s integral formula, calculate the following integrals:
, C : |z| = 2.
(i) C (9−zz2dz
∫ sin πz2)(z+i)
+cos πz 2
(ii) C (z−1)(z−2) dz, C : |z| = 3.
∫ ezt
1
(iii) 2πi
dz, t > 0, C : |z| = 3.
∫ Cdzz2 +1
(iv) C z(z+πi) , C : |z + 3i| = 1.
∫
(v) C (z+1)z−1
dz, C : |z − i| = 2.
2
∫ z2 −z−1(z−2)
(vi) C z(z−i)2 dz, C : |z − 2i | = 1.
∫
(vii) C (z2dz
, C : |z − i| = 2.
+4)2
∫ cosh(πz)
(viii) C z(z2 +1) dz, C : |z| = 2.
15. Evaluate
∫ dzthe following integrals:
(i) C 2−z
, C : |z| = 1.
∫ eaz
(ii) C z−πi dz, where C is the ellipse |z − 2| + |z + 2| = 6.
∫
ez
(iii) C z2 (z+1)
C : |z| = 2.
3 dz,
∫
z−3
(iv) C z2 +2z+5 dz, C : |z + 1 − i| = 2.
2