Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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Review
1. Compute the cubic roots of the number 1 + i.
2. Identify R2 with C. Write down the complex function, z 7→ f (z), corresponding to the following map from R2 to R2
x
2x + 3y
=
.
y
x + 0.5y
Write down the function u, v such that f = u + iv.
3. Prove the following statements.
(a) A real linear map f : R2 → R2 is complex linear if and only if f (i) =
if (1).
(b) Let a, b, c, d ∈ R and
x
a b
x
T
=
.
y
c d
y
There exists a unique pair of complex numbers λ and µ such that
T (z) = λz + µz̄,
(c) The map T (z) = λz + µz̄ is complex linear if and only if µ = 0.
By a region, we mean an open connected subset of C.
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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2
Example Sheet 1: Complex Differentiation
1. Let r be a non-zero real number, c a real number and k ∈ C, satisfying the
relation |k|2 > cr. Prove that the equation
r|z|2 − k̄z − kz̄ + c = 0.
represents a circle. Determine its centre and radius.
2. Let c ∈ R and k ∈ C. Prove that k̄z + kz̄ + c = 0 represents a straight line.
3. let f : C \ {0} → C \ {0} be the inversion f (z) = z1 . Prove that f takes a
circle to a circle or a line.
4. Suppose that f : U → C is complex differentiable where U is a region.
Define Ū := {z : z̄ ∈ U }. Prove that g : Ū → C given by the formula
g(z) = f (z̄)
is complex differentiable. Write g 0 in terms of f .
5. Let log z := log |z| + i arg z. Use the Cauchy-Riemann equation to prove
that the logarithm function log : C \ (−∞, 0] → C is holomorphic.
6. Let f : {a + ib : a > 0} → C \ (−∞, 0] be the function z 7→ z 2 . Prove that
f −1 is holomorphic.
7. Determine at which points the following functions are differentiable:
zRe(z),
z̄
,
|z|2
z z̄.
8. Find all holomorphic functions f whose real part is u(x, y) = 2xy + 2x.
9. Let us consider a function f in polar coordinate (r, θ). Let U be a domain
on which f (r, θ) = ũ(r, θ) + iṽ(r, θ) is defined. Prove the polar form of the
Cauchy-Riemann equation :
1
∂r ũ = ∂θ ṽ,
r
1
∂r ṽ = − ∂θ ũ.
r
10. If F 0 (z) = f (z) in a region D we say F is a primitive of f . Prove that f (z) =
|z|2 does not have a primitive. (Hint: Use Cauchy-Riemann equation.)
3
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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Example Sheet 2. Conformal Map. and Möbius Transforms
1. Let r be a positive number and a, b ∈ C. Prove that the equation
determines a circle or a line.
|z−a|
|z−b|
=r
2. Write down the Möbius transform that takes (0, i, 1) to (0, 1, −1).
3. Let b be a complex number with |b| < 1. Let
f (z) =
−z + b
.
−b̄z + 1
(a) Prove that f maps the unit disc D = {z : |z| < 1} to itself.
(b) Prove that |f (z)| = 1 if |z| = 1
(c) Give a formula for f −1 .
(d) Prove that f : D → D is bijective.
4. Denote by SL2 the family of special Möbius transforms
az + b
: a, b, c, d ∈ R, ad − bc = 1 .
SL2 =
cz + d
Let H = {x + iy : y > 0} denote the upper half plane. Prove that
(a) Each map from SL2 takes H to H.
(b) For any two points z, w ∈ H there exist a map from SL2 taking z to
w.
5. Compute the Stereographic distance between 2i and ∞.
6. Prove that if z ∈ C∗ corresponds to a point (X, Y, Z) in S 2 , then − z̄1 corresponds to its antipodal point.
7. Prove that if z1 , z2 , z3 , z4 are distinct points in C then they lie in a circleline
if and only if the cross ration [z1 , z2 , z3 , z4 ] is a real number.
Hint to (4b): First find a map taking z to i. Try d = 0. Solve the equation
f (z0 ) = i and take care of both the real and the imaginary parts.
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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4
Example Sheet 3: Power Series and Contour Integral
P
k
1. If we know ∞
k=0 ak (z − i) converge for z = 4 and diverges for z = −8,
what can you say about the convergence of the following power series?
∞
X
∞
X
n
an (1 + i) ,
n=0
n
an 9 ,
∞
X
n=0
2. Prove that the power series function f (z) =
disc D of convergence.
(−1)n an 6n .
n=0
P∞
3. Find an analytic continuous of the power series
R1
4. Compute 0 f (t) dt where f (t) = t + 2i.
n=0
an z n is analytic in its
P∞
n=0 (1
− z)n .
5. Write down a parametrisation for the triangle, with vertices the origin, R
and R + iR, and for each side of the triangle.
6. Let γ be the upper half of the unit circle centred
√ at the origin. Use the
following two parametrisations z(t) = −t + i 1 − t2 , −1 < t < 1 and
z̃(t) = cos(t) + i sin(t), t ∈ (0, π) of γ to compute the integral
Z
1
dz.
γ z
(Please do not use any theorems)
R
7. Compute |z|=1 z̄dz and conclude that f (z) = z̄ does not have a primitive
on {z : |z| < 2}.
2
8. Let f (z) = e−z . Is f a holomorphic function?
(a) For R > 0, prove that
Z
1
f (z)dz ≤ .
R
[R,R+iR]
(b) Prove that
√
π
f (z)dz =
.
2
[0,R]
Z
lim R→∞
(c) Use Goursat’s theorem to prove that
√
Z ∞
Z ∞
π
2
2
.
cos(2t )dt =
sin(2t )dt =
4
0
0
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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5
Example Sheet 4. Primitives and Caychy’s Integral formula
R
1. Calculate γ sin(z)dz where γ = {t + it : 0 ≤ t ≤ 1}, firstly by the
definition, then by finding a primitive.
2. For each of the following functions, calculate the indicated integrals without
using Cauchy’s theorems. Hint overleaf.
R
2
(a) γi zez dz where γ1 = [1+i, i] and γ2 is the locus of |z−a|+|z−b| = r
where a, b, ∈ C, r ∈ R and r > |a − b|.
R
(b) γ (z2z+1) dz if γ the locus of |z − 27| + |z − 26| = 2.
3. Let γ be a smooth curve with parameterization z : [a, b] → C and φ a
holomorphic function. Denote by γ̃ the image of γ by φ : γ̃ = φ(γ). Prove
that if f is a continuous function then
Z
Z
f (z)dz = f ◦ φ(z) φ0 (z)dz.
γ
φ(γ)
4. If f is an even continuous function, i.e. f (z) = f (−z), prove that for any
r > 0,
Z
f (z)dz = 0.
|z|=r
P∞
n
5. Let f (z) =
n=0 an (z − z0 ) with radius of convergence R > 0. Let
r ∈ (0, R). Prove that
Z
1
f (z)
dz.
ak =
2πi |z−z0 |=r (z − z0 )(k+1)
Hint overleaf.
6. Let z1 , z2 ∈ C and r1 , r2 be two real numbers such that r1 + |z1 − z2 | < r2 .
(The circle |z − z1 | = r1 is in the interior of the larger circle |z − z2 | = r2 .)
Suppose there exists R1 < 12 r1 and R2 > r2 such that f is holomorhpic in
the region
U = {|z − z1 | > R1 } ∩ {|z − z2 | < R2 }.
Prove that
Z
Z
f (z)dz =
|z−z1 |=r1
in two ways.
f (z)dz
|z−z2 |=r2
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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(1) Run a vertical and a horizontal lines through z1 giving us four closed
simple curves γi , such that
Z
Z
XZ
f (z)dz =
f (z)dz −
f (z)dz.
i
|z−z2 |=r2
γi
|z−z1 |=r1
For each γi there is a line segment sliding the larger disc into two regions,
one of which contains γi and on which f is holomorphic.
(2) Divide the region U into two simply connected regions with boundary
curves Γ1 and Γ2 satisfying
Z
Z
Z
Z
f (z)dz +
f (z)dz =
f (z)dz −
f (z)dz.
Γ1
7. Calculate
|z| = 1.
|z−z2 |=r2
Γ2
R
1
dz
|z|=1 (z−a)(z−b)
Alternative : calculate
r 6= 3.
where a, b ∈ C and a, b do not line on the circle
1
dz,
γ (z−2)(z−3)
R
|z−z1 |=r1
γ is a circle |z − z0 | = r where r 6= 2
8. Use Cauchy’s formula to prove that, for C the circle |z| = 1 and a, b complex numbers with |a| > 1 and |b| < 1,
Z
1
2πi
dz =
.
3
(b − a)3
C (z − a) (z − b)
Hints.
Q 2 (b). |z − a| + |z − b| = r, for r > |b − a|, is an ellipse with foci a and b.
.
Q4. Write down an explicit power series expansion f for (z−zf0(z)
)(k+1)
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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7
Example Sheet 5: Cauchy’s formula, derivative formulas,
applications
1. Prove the following extension to Louville’s theorem. If f is an entire function with limz→∞ f (z)
= 0, then f is a constant.
z
2. Compute the following Integrals.
Z
Z
ez+1
1
dz,
dz,
2
|z|=1 z
|z|=2 z + 1
3. Compute
R
|z|=2
Z
|z|=0.5
1
dz,
z2 + 1
Z
[− 12 , 21 ]
z
dz.
z2 + 1
z n (1 − z)−m dz where n, m ∈ N .
4. Denote by γ1 the segment from 1 to r > 1 on the real axis, γ2 the arc from r
to w = reiθ along the circle centred at 0 of radius r. Here θ ∈ (0, π2 ). Prove
that
Z
1
dz = ln w,
γ1 ∪γ2 z
Recall ln w = ln r + iθ.
R
5. Let and a be positive numbers. Calculate R z1 dz where R is the rectangular with vertices at −a − i, a − i, −a + i, and a + i by the definition
of curve integrals. (I am aware you know a quicker way to calculate this.)
Prove that
Z
Z
1
1
= πi, lim
= πi.
lim
→0 [a+i,−a+i] z
→0 [−a−i,a−i] z
6. Suppose that f is holomorphic in a region containing {|z − z0 | < 2} with
f (z0 ) = 0, f 0 (z0 ) 6= 0, f (z) 6= 0 for z 6= z0 . Use Cauchy’s theorem to prove
that
Z
1
2πi
dz = 0
.
f (z0 )
|z−z0 |=1 f (z)
7. Prove that the function
1
sin z
has singularities at nπ where n ∈ Z.
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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Example sheet 6.
1. Prove that if an entire function f satisfies |f (z)| ≤ c|z|n for some natural
number n, then f is a polynomial of degree at most n.
2. Prove that if f is a non-constant entire function then f (C) is dense in C.
P
1
z
z log n
.
3. Denote ζ(z) = ∞
n=1 nz the Riemann ζ function. Recall that n = e
P
1
• If Re(z0 ) > 1, prove N
n=1 nz converges uniformly on a disc D(z0 , δ).
• Prove that ζ is analytic on {z : Re(z) > 1}.
4. Let f be a holomorphic function on a deleted disc D0 (z0 , r) = {z : 0 <
|z − z0 | < r}. Show that the following statements are equivalent :
(a) ord(f ; z0 ) = n;
(b) there exists positive numbers c1 , c2 and δ such that
c1 |z − z0 |n ≤ |f (z)| ≤ c2 |z − z0 |n ,
0 < |z − z0 | < δ.
5. Suppose that both f and g have singularity at z0 of finite orders. Prove that
ord(f g; z0 ) = ord(f ; z0 ) + ord(g; z0 ),
1
ord( ; z0 ) = − ord(g; z0 ).
g
6. Let f be a non-constant holomorphic function in a region U . Let V ⊂ U be
a closed bounded subset of U . Then for any w ∈ C, {z ∈ V : f (z) = w}
has a finite number of elements or f is a constant on V .
P∞
P
n
7. Given two power functions f (z) = ∞
n=0 bn (z−
n=0 an (z−z0 ) and g(z) =
n
z0 ) on {|z − z0 | < R}, suppose that there exists a sequence of numbers
{zk } such that f (zk ) = g(zk ) and limk→∞ zk = z0 . Prove by induction that
an = bn for all n.
P
n
8. Suppose that f (z) = ∞
n=−∞ an (z − z0 ) for R1 < |z − z0 | < R2 . Prove
that for all r ∈ (R1 , R2 ),
Z
1
f (z)
an =
dz.
2πi C(z0 ,r) (z − z0 )n+1
9. Let fn be a sequence of holomorphic functions on U = {|z − z0 | < 3r}.
Suppose that there exists a dense subset V of U on which fn converges.
Suppose that for a positive number M , |fn (z)| ≤ M for all n and all z in
{z : |z −z0 | ≤ 2r}. Prove that fn converges uniformly on {z : |z −z0 | ≤ r}
to a holomorphic function.
9
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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Example Sheet 7
1. Show
P∞by an example the statement that a−n + · · · + an → s is not the same
as n=−∞ an = s.
2. Deduce the Laurent series for f (z) =
(a). D(0, 1);
1
(z−2)(z−3)
in the following regions.
(b). 2 < |z| < 3;
(c). |z| > 3.
3. The following functions have singularities at 0. Describe the type of the
singularity for each function.
3+z
;
+ 3z 2 )
z 4 (5
1
,
tan z
sin z
;
z
1
cos( );
z
ez
z
−1−z−
z2
2
;
4. If the order of the pole at z0 is n, prove that
Res(f ; z0 ) = lim
z→z0
1
d
( )n−1 (z − z0 )n f (z).
(n − 1)! dz
5. Evaluate the residues of the following functions at their singularities:
1
,
1 + z2
6. Is sin(z) bounded on C?
1
,
sin z
(
z+1 2
)
z−1
1
ez .
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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10
Example Sheet 8.
1.
(a) State Cauchy’s theorem for a closed curve γ ≈ 0.
(b) Prove the following Cauchy’s theorem for
R a simply connected domain:
If U is a simply connected region then γ f (z)dz = 0 for every holomorphic function f : U → C and every closed piecewise C 1 curve γ
in U .
(c) What can you conclude for a star region?
2. How many zeros, counting multiplicity, does the function f (z) = z 7 −3z 3 +
ez + 7 have in the annulus A = {z : 1 < |z| < 2}?
3. Hurwitz’s theorem. Let U be a region, fn : U → C a sequence of holomorphic functions converging to a function f uniformly on compact subsets of
U . If each fn has no zero on U prove that f has no zero in U unless f is
identically zero.
rz
e
4. Let f (z) = 1+e
z where r ∈ (0, 1) is a real number and R a positive number. Let γ be the rectangular, oriented positively, with one side the segment
[−R, R] on the real axis and a parallel side passing through 2πi. Find the
poles of f inside the rectangular and compute their residues.
5. Prove that
Z
∞
−∞
π
erx
dx =
.
x
1+e
sin(πr)
6. Optional Question. Let γ be a piecewise smooth curve and f is defined and
continuous on γ. For m ≥ 1 define for z 6∈ {γ},
Z
f (ζ)
gm (z) =
dζ.
m
γ (ζ − z)
0
(z) = mgm+1 (z). Hint. Use
Prove that g is holomorphic on C \ {γ} and gm
the definition of complex differentiation.
Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri
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Example Sheet 9.
Let D = {z : |z| = 1}.
1.
(a) Prove Schwartz’s Lemma. If f : D → D is a holomorphic function
with f (0) = 0, then either (1) f (z) = cz for a complex number c with
|c| = 1 or (2) |f (z)| < |z| for all z ∈ {z : 0 < |z| < 1}.
(b) Prove that every bi-holomorphic function f : D → D with f (0) = 0
is of the form f (z) = cz where |c| = 1.
(c) Let b be such that |b| < 1. Prove that Gb : D → D given by the
z−b
formula Gb (z) = 1−
is bi-holomorphic.
b̄z
(d) Prove every bi-holomorphic mapping from D onto D is of the form
cGb for some c, b ∈ C such that |c| = 1 and |b| < 1.
(e) Let U be a simply connected region which is not the whole plane and
z0 ∈ U . Suppose that f, g : U → D are bi-holomorphic functions
such that f (z0 ) = g(z0 ) = 0, f 0 (z0 ) > 0 and g 0 (z0 ) > 0. Prove that
f = g. (Proposition 9.3.2)
2. Suppose that f is an entire function with no zero and for r > 0 define
m(r) = inf |z|=r {|f (z)|}. Prove that m is non-increasing.
3. Prove that there is a homeomorphism (continuous map with continuous inz
is a
verse) from any simply connected region to C. Hint. g(z) = 1−|z|
homeomorphism from D to C.
4.
(a) Let f be an entire function that is not a polynomial. Given w ∈ C
prove that for any > 0 and R > 0 there exists z with |z| > R such
that |f (z) − w| < . Hint. Investigate the singularity of g(z) = f ( z1 ).
(b) If f is a one-to one entire function, then f (z) = az + b for some
complex numbers a, b ∈ C.
5. Suppose that f is a nowhere vanishing holomorphic function in a star region
U . Prove that there exists a holomorphic function g s.t. eg(z) = f (z).
6. Let U be a region, fn : U → C a sequence of holomorphic functions converging to a non-constant function f uniformly on compact sets. Prove
(1) Hurwitz’s Theorem. Let z0 ∈ U and w0 = f (z0 ). For any > 0 sufficiently small there is an integer N () > 0 such that for each n ≥ N (),
there exists zn with |zn − z0 | < and fn (zn ) = w0 .
(2) If each fn is one to one then so is f .