Complex Analysis I MT333P
Problems/Homework
Recommended Reading:
Bak, Newman: Complex Analysis, Springer
Conway: Functions of One Complex Variable, Springer
Ahlfors: Complex Analysis, McGraw-Hill
Jaenich: Funktionentheorie, Springer
Arnold: Complex Analysis, http://www.ima.umn.edu/~arnold/502.s97/complex.pdf
1. Compute the Jacobi matrix of the map
f : R3 → R2
2
with f (x, y, z) = x + y + xz, ex −xy .
Solution:
1+z
2
(2x − y)ex −xy
d(x,y,z) f =
1
2
−xex
x
0
−xy
2. Compute
Z t
d x34 + x3 + 9 dx .
dt t=1 −t
Solution: 20
3. Compute the limits
lim x3 e−x
,
x→∞
lim x log(x) .
x→0
Solution: Both limits are 0.
please hand up 1-3 Monday, 29/9 in class
4. Express the following in the form a + ib with a, b ∈ R:
(3 + 7i)(3 + 3i)
,
(3 + i)2
17i + 1
,
cos(i) .
Solution: (3 + 7i)(3 + 3i) = 9 − 21 + i(9 + 21) = −12 + 30i,
(3 + i)2
(3 + i)2 (1 − 17i)
(3 + i)2 (1 − 17i)
110 − 130i
=
=
=
2
17i + 1
(17i + 1)(1 − 17i)
1 + 17
290
cos(i) =
eii + e−ii
1
e
=
+ = cosh(1)
2
2e 2
5. Compute modulus and an argument for
1+i
,
2 + 3i
,
√
1
3
−i
.
2
2
Solution:
1+i=
√
π
2 exp i
4
,
2 + 3i =
√
3
13 exp i arctan
2
,
√
1
3
−π
−i
= exp i
2
2
3
6. Write the polynomial p(x) = x4 + 4 ∈ R[x] as a product of quadratic/linear real polynomials.
2
Solution: Abbreviating j := exp( iπ
4 ), hence j = i we compute
√
√
√
√
x4 + 4 = (x2 − 2i)(x2 + 2i) = (x + 2j)(x − 2j)(x + 2ij)(x − 2ij)
√
√
√
√
= (x + 2j)(x − 2ij) × (x + 2ij)(x − 2j)
= (x2 + 2x + 2) × (x2 − 2x + 2)
please hand up 4-6 Monday, 6/10 in class
7. For the following functions u : U ⊂ C → R find functions v : U → R such that f = u + iv is holomorphic on U :
u(x + iy) = x + x2 − y 2 , U = C
and
Solution: v(x, y) = y + 2xy
u(x + iy) = ln(x2 + y 2 ) , U = {x + iy ∈ C | x, y ∈ R, x > 0}
Solution: v(x, y) = arctan(y/x)
8. Determine all harmonic two variable polynomials p(x, y) ∈ R[x, y] of degree ≤ 3.
Hint: A polynomial p(x, y) ∈ R[x, y] of degree ≤ 3 has the form
X
p(x, y) =
pi,j xi y j ,
i,j≥0, i+j≤3
in total 10 coefficients pi,j .
9. Describe the sets
ex+iy | x, y ∈ R, 0 < x < 1, 0 < y < π
and
π
3π
ex+iy | x, y ∈ R, 0 < x < 1, < y <
4
4
by inequalities.
Solution: {z ∈ C | 1 < |z| < e, =(z) > 0}, {z ∈ C | 1 < |z| < e, =(z) > |<(z)|}
please hand up 7-9 Monday, 13/10 in class
10. Let f : C → C be the function with f (z) = z for all z ∈ C. Compute
γ = γi :
R
γ
f (z) dz =
R
γ
z̄ dz for the following curves
(a) γ1 : [0, 2π], γ1 (t) = eit ,

:0≤t≤1
 t
1 + i(t − 1)
:1≤t≤2 ,
(b) γ2 : [0, 3], γ2 (t) =

3 − t + i(3 − t) : 2 ≤ t ≤ 3
(c) γ3 : [0, 1], γ3 (t) = z + tv for some z, v ∈ C. More generally prove that
R
γ
z̄ dz is imaginary whenever γ is closed.
9
(d) γ4 : [−1, 1], γ4 (t) = |t| + i(t − t).
R
Hint: If you can say (guess from 10a, 10b, 10c) what γ z̄ dz means “geometrically” for a closed curve γ, you
need not rigorously prove your answer to 10d.
Solution: 2πi, i, z̄v, 8i/5. The integral is 2i times the area of the interior of the curve.
11. Let U ⊂ C and γ, µ : [0, 2] → U be smooth curves given by
U = {z ∈ C | <(z) ≥ 0, =(z) ≥ 0, |z| ≥ 1}
and γ(t) = 1 + t, µ(t) = i + it.
Find a smooth map H : [0, 1] × [0, 1] → U such that H(0, t) = γ(t) and H(1, t) = µ(t).
Solution: H(s, t) = (1 + t)e
iπs
4
.
12. Describe the following regions in C by inequalities for =(z), <(z), |z|:
(a) U1 = {(x + iy)2 | x, y ∈ R+ },
(b) U2 = {sin(x + iy) | x, y ∈ R},
1
(c) U3 = { x+iy
| x, y ∈ R, x, y > 1}.
13. The lenght of a C 1 -curve γ : [a, b] → Rn is defined as
Z
length(γ) =
b
kγ 0 (t)k dt .
a
(a) For λ ∈ R, r, b > 0, compute the length of the helix γ : [0, b] → R3 , γ(t) = (r cos(t), r sin(t), λt).
(b) Prove that if γ[a, b] → U ⊂ C is C 1 then
Z
f (z) dz ≤ length(γ) max{|f (z)| | z ∈ U }
γ
for any continuous function f : U → C.
(c) Assume that f is a holomorphic function on B1 (0) with f (0) = 0 and f 0 (0) 6= 0. Prove that for sufficiently small
r > 0 we have
I
1
2πi
dz = 0
.
f (0)
|z|=r f (z)
Hint: Prove first that 1/f is holomorphic on Br (0)\{0} for sufficiently small r and then that the integral does not
depend on r provided is is small. You might use that f (z) = f 0 (0)z + q(z) with some q such that limz→0 q(z)
z = 0
exists.
Solution:
I
I
Z 2π
Z 2π
1
1
ireit
ireit
dz = lim
dz = lim
dt
=
lim
dt
r→0 |z|=r f (z)
r→0 0
r→0 0
f (reit )
f 0 (0)reit + q(reit )
|z|=r f (z)
Z 2π
Z 2π
2πi
i
i
= lim
dt
=
dt = 0
.
it
it
r→0 0
f
(0)
q(re
)
q(re
)
0
0
0
f (0) +
f (0) + lim
r→0 reit
reit
14. Let V be a vector space over R with scalar product denoted by h· | ·i. The norm of a vector v ∈ V is the real number
p
kvk = hv | vi
and the angle between two vectors v, w ∈ V \ {0} is defined to be the number ∠(v, w) ∈ [0, π] such that
cos ∠(v, w) =
hv | wi
.
kvkkwk
A linear map A ∈ Hom(V, V ) is called conformal if it preserves angles, i.e. if
∠(Av, Aw) = ∠(v, w)
v, w ∈ V \ {0} .
for all
2
(a) Let V = R with the standard scalar product. Determine all conformal 2 × 2-matrices
a
c
b
d
∈ Hom(R2 , R2 ).
(b) Let A ∈ Hom(V, V ) be Hermitian, i.e.
hAv | wi = hv | Awi for all
v, w ∈ V ,
and assume that A is conformal. How many different eigenvalues can A have?
15. Let U ⊂ C be open and f : U → C be holomorphic. Prove that the map
g : U := {z | z ∈ U } → C
, z 7→ f (z)
is holomorphic.
Solution:
lim
z→z0
f (z) − f (z)
f (z) − f (z0 )
f (z) − f (z0 )
f (u) − f (z0 )
= lim
= lim
= lim
= f 0 (z0 )
z→z0
z→z0
u→z0
z − z0
z − z0
z − z0
u − z0
16. Assume r > 0 and that the complex numbers ak , bk , k ∈ N0 are such that the series
for all z ∈ Br (0) to the same number, i.e
f (z) :=
∞
X
ak z k =
k=0
∞
X
P∞
k=0
ak z k ,
P∞
k=0 bk z
k
converge
bk z k =: g(z)
k=0
for all z ∈ Br (0).
(a) Prove that ak = bk for all k ∈ N0 .
Solution: ak = f (k) (0)/k! = g (k) (0)/k! = bk .
1
(b) Let f : U = C \ {i, −i} → C be the function with f (z) = e 1+z2 for all z ∈ U . What is the radius of convergence
of the Taylor series
∞
X
f (k)
(z − 7 + 2i)k
k!
k=0
of f at 7 − 2i?
√
Solution: The maximal r such that f is holomorphic on Br (7−2i) is r = min{|7−2i−i|, |7−2i+i|} = |7−i| = 50.
please hand up 10-16 Wednesday, 12/11 in class
17. Let f : B1/10 (1/4) → C be the function given by
f (z) =
z2 +
1
5z
z2 + z
10100
+ 1000000
(z 3 + z)
for all
z ∈ B1/10 (1/4) .
What is the radius of convergence of the Taylor series of f at 1/4?
18. Compute the Taylor series at 0 of the function f given by the formula
f (z) =
z2 + 1
.
z 3 + 18
Hint: You need to give a formula for ak ∈ C such that f (z) =
P∞
k=0
ak z k .
19. Compute the first four terms of the Taylor series at 0 of the function f with f (z) = sin(ez ).
please hand up 17-19 Wednesday, 26/11 in class
20. Find the Laurent series of the function
f : C \ {1, −1} → C
around 0 which converges near 2.
with f (z) =
z
z 2 − 4z + 3
for all
z ∈ C \ {1, −1}
21. Classify the (isolated) singularities of the following functions fi as removable, pole, essential. Also give the order of
the poles.
(a) f : C \ {−1, 1} → C, f (z) =
z 3 + z + 34
,
(z 2 − 1)3
cos(z) sin(z 2 )
,
z5
z
(c) f : C \ {i, −i} → C, f (z) = sin
.
z2 + 1
(b) f : C \ {0} → C, f (z) =
22. Let f : C → C be holomorphic. Assume that
f (z)| ≤ |z|7
z ∈ C and f (7) (0) = 7!i .
for all
Compute f (−i).
please hand up 20-22 Monday, 8/12 in class
23. Prove that if sin(z) = 0 for some z ∈ C, then z/π ∈ Z.
Hint: sin(z) = eiz − e−iz /2i
24. Classify the (isolated) singularities of the following functions fi as removable, pole, essential. Also give the order of
the poles.
(a) f : C \ Z → C, f (z) =
z2
,
sin(z)3
2
ez − 1
,
(b) f : C \ {−1, 0, 1} → C, f (z) = 2 2
z (z − 1)
1
(c) f : C \ {0, kπ
| k ∈ Z \ {0}} → C, f (z) =
1
.
sin(1/z)
25. Determine all meromorphic functions f : C → C with |f (z)| < e1/|z| − 1 for all z ∈ C, where f is defined.
Hint: What singularities can f have? You might consider looking at the Laurent series.
26. (a) Prove that a non-constant meromorphic function f : C → C has at most finitely many poles and zeros in B1 (0).
(b) Let f : C → C be a meromorphic function and assume that the function f˜: C → C with f˜(z) = f (1/z) for all
z ∈ C, where f (1/z) is defined, is also meromorphic. Prove that f is a rational function, i.e. that there are
p(z), q(z) ∈ C[z] such that f (z) = p(z)/q(z) for all z ∈ C which are not a pole of f .
Hint: The function 1/z maps B1 (0) to C ∪ {∞} \ B1 (0). In C, a bounded sequence has a convergent subsequence.
please hand up 23-26 Monday, 15/12 in class
27. Write the following complex numbers z in the form z = a + ib and z = reiφ with suitable real numbers a, b, r > 0, φ.
z=
1+i
1−i
, z = (2 + 3i)4
and z = e1+i .
28. Find the Laurent series expansion around 0 of the function
f : C \ B2 (0) → C
, f (z) =
z2
(z 3 + 1)(z − 1)
for all
z ∈ C \ B2 (0) ,
which converges in a neighbourhood of 88.
29. Find a function v : C → R such that f = u + iv : C → C is holomorphic, where u : C → R is the function with
4
u(z) = (<(z)) for all z ∈ C.
30. The following formulas give a meromorphic function f : C → C. Find all poles of f and determine their order.
f (z) =
sin(z)
z2 − 4
, f (z) =
sin(z)
(cos(z))3
, f (z) =
sin(z)
.
z4 − π4
31. Decide which of the isolated singularities of the following functions f : C \ S → C are removable singularities, poles,
essential singularities.
1
S = {−2, 2} , f (z) = sin
(z 2 − 4)
2
S = {−1, 1}
ez −1 − 1
, f (z) = 2
(z − 1)(z + 1)
32. Assume that the series
P
k∈Z
ak (3 − i)k converges and that
X
k∈Z
ak (z − i)k =
1
sin(z)
for all
z ∈ B1/888 (3) .
Find the smallest possible r ∈ R, r > 0 and the largest possible R ∈ R such that
z ∈ C with r < |z − i| < R.
2
P
k∈Z
ak (z − i)k converges for all
33. Determine all meromorphic functions f : C → C such that |f (z)| ≤ e−|z| for all z ∈ C with |z| > 1.