Complexe Functies

Complexe Functies spring 2010

Schedule (rooster)
The class WISB311 is held on Mondays, from 9:00 to 11:00, in the room 205 of the Buys Ballot building. The exercise class are Mondays, from 11:00 to 13:00, in the same room.
Teacher  The teacher of this class is André Henriques, and the teaching assistent is Leslie Molag (l.d.molag⊗students.uu.nl).
The book  We shall be following the book Complex analysis of Serge Lang.
The exercises  This class has mandatory exercises that count for 10% of the final grade.
The exam   There will be a written exam at the end of the 3rd block, and an oral exam on June 21st and 22nd, in room 610 of the wiskundegebouw.
Each exam counts for 45% of the final grade.
Hertentamen op maandag 23 Augustus
Time: from 09.00 to 12.00
Location: BBL 005.

The material

week Material covered so far Exercises (Solutions written by Leslie)
Hand in exercise.
(To hand in on week n+1)
6 The e z and log(z) functions. Analytic functions. Holomorphic functions.
The Cauchy Riemann equations. Angles under holomorphic maps.		 I§1: 4, 5, 7, 8. I§2: 2, 8, 10
I§2: 13 (p. 12)
7 Convergence radius. Analyticity of functions defined via power series.
Differentiation of power series. Analytic ⇒ Holomorphic.		 I§2: 3, 4, 11, 12. I§3: 1, 2, 3, 4.
II§2: 8 (p. 59)
8 Formulas for the convergence radius (p. 55 & 57).
The set of zeroes of an analyticity function is discrete. Analytic continuation.		 I§4: 1, 3, 5. II§1: 1, 3, 4. II§2: 1, 3, 4, 5, 6, 10.
II§3: 2 (p. 68)
9 Inverse function of an analytic function (p. 76 - 79).
My proof used Holomorphic ⇒ Analytic, which we haven't shown yet.		 II§1: 6, Prove the cases 0 and ∞ of Thm 2.6 (p. 55).
II§2: 7, 11, 12, 13, II§4: 1, 2.
Hand in
10 Open mapping theorem. Maximum modulus principle. Integrals along paths. II§3: 4, 5 II§5: 1, 2, 3, 4, 5, 6 II§6: 1 - 5
II§6: 6 (p. 83)
11
Hertentamen week
12 Invariance of the integral under homotopies of paths and of loops.
The concept of simply connected topological space.		 III§1: 2, 3. III§2: 1, 3, 4, 8, 10, 11.
III§2: 7 (p. 103)
13 Winding number. Existence of local primitives. Holomorphic ⇒ Analytic. III§5: 1, 2, 3, 4. III§6: 1, 2, 3, 4, 5, 6, 7, 8.
Hand in
14
Paasdag
15 Poles and residues. The residue theorem (see pages 173,174, and 144). III§7: 1, 3. V§2: 6, 11, 13.
VI§1: 1, 3, 5, 7, 9, 11, 13, 19, 20.
III§7: 2 (p. 132) write
the proof of Corollary 7.6*.
16
-------------------------------- Het deeltentamen vindt op maandag 19 april plaats, van 9-12 uur in zaal BBL 161.
*sorry, I skrewed up. I should
have written Theorem 7.7...
You may chose which one
you hand in: either 7.6 or 7.7.
17 Proof of the residue theorem (p. 143-154).
 V§1: 1, 2. V§2: 1, 2, 3, 4, 5, 8.
18 Existence of Laurent expansions. V§1: 6, 7. V§2: 7, 9, 10, 12, 15.
V§2: 14 (p. 165)
19 Uniform limits of analytic functions. The function Σn ∈ ℤ (z-n)-1. Liouville's thm. V§1: 3, 4, 5. V§3: 1, 2, 3, 4, 5.
Exercise A (below)
20 Computation of definite integrals. The example of ∫-∞∞ cos(πx/2)/(1+x4) dx. V§3: 9, 10, 11. VI§2: 1, 2, 3, 4.
Exercise B (below)
21 More definite integrals: the Mellin transofrm of 1/(1+x2). The Riemann sphere ℂℙ1.
Classification of holomorphic maps ℂℙ1 → ℂℙ1. Fractional linear transofrmations.		 VI§2: 5, 6, 7, 8, 9, 11, 17, 18, 19, 20.
VI§2: 10 (p. 206)
22 Automorphisms of the Riemann sphere. Analytic automorphisms of the
upper half plane and of the unit disc. Statement of the Riemann mapping theorem.		 VII§5:1, 2, 3, 4, 5, 6.
VII§4: 7 (p. 230)
23
Geen hoorcollege, wel werkcolege
24
Het mondeling eindtentamen vindt op maandag en diensdag 21 en 22 Juni plaats, in zaal 610 van het wiskundegebouw.


Exercise A:
Given a curve K in the complex place, a path γ:[0,1]→ℂ is called transverse to K if the set S := {t ∈ [0,1] | γ(t) ∈ K} is finite, does not contain the points 0 and 1, and for every element t ∈ S, the derivative γ'(t) is not parallel to the tangent line of K at γ(t). Similarly, a chain c = Σj njγj is called transverse to K if all the paths γj are transverse to K.
Let c = Σj njγj be a closed chain in ℂ\{0}, and let us assume that c is transverse to the positive real axis ℝ>0. For every point t ∈ Sj := {t ∈ [0,1] | γj(t) ∈ ℝ>0}, define εj(t) to be +1 if the imaginary part of γ'j(t) is positive, and -1 otherwise. Recall the the winding number W(c) is defined to be the integral over c of the function 1/(2πiz).
Given the above definitions, prove that the following formula holds:
W(c) = Σj Σ t∈ Sj  εj(t) nj

Hint: Use the logarithm function.

Exercise B:
Given a function f on the Real line, its Fourier transform is the function given by f^(ξ) := ∫ℝ f(x)e-2πixξ dx.
Let f(x):= sech(x) = 1/cosh(x). Compute f^(1) with the methods of Theorem 2.2 (page 194).
(Pay attention to the convergence arguments!)
Description of the final oral exam:
Each student will have 20 minutes to prepare (in the back of the room), and 20 minutes to present the question at the blackboard.
The question will consist of two parts: 10 minutes for an item of the folowing list, and 10 minutes for a werkcollege opgave (not an inlerveropgave).

• Which functions may be called "the complex logarithm function"? Prove that no such function can be defined on C*.
• Prove that a function satisfies the Cauchy Riemann equations if and only if it is complex differentiable.
• Prove that holomorphic functions are caracterized by the property of preserving angles.
• What is the convergence radius of a power series? Prove that ρ = limsup |an|1/n.
• Prove that a convergent power series defines an analytic function.
• Prove that the zero-set of an analytic function is discrete.
• What is analytic continuation? Prove the uniqueness of analytic continuation.
• Sketch the proof that an analytic function with non-zero derivative defines a local isomorphism.
• State and prove the open mapping theorem.
• State and prove the maximum modulus principle.
• Prove that integrals along paths are invariant under homotopies fixing the end points.
• Prove that integrals along loops are invariant under homotopies.
• Prove that every holomorphic function on a simply connected domain has a primitive.
• Prove "Holomorphic => Analytic".
• State the residue theorem and sketch its proof.
• Sketch the proof that every holomorphic function defined on an anulus has a Laurent expansion.
• Prove that a uniform limit of analytic functions is analytic.
• What does it mean for a map CP1→CP1 to be holomorphic? Classify all such maps.
• Prove that every (holomorphic) automorphism of the Riemann sphere is given by a fractional linear transformations.
• Prove that all (holomorphic) automorphisms of the upper half plane are of the form f(z) = (az+b)/(cz+d) with a,b,c,d ∈ R and ad-bc > 0.


Location of the exam: Wiskundegebouw, zaal 610.

Shedule of the oral exam (this is now definitive)
Ackermann, Arno: maandag 21 Juni om 14:40
Bongers, Stephan: maandag 21 Juni om 13:50
Boven, Hasse van: maandag 21 Juni om 13:25
Dokter, Kasper: diensdag 22 Juni om 13:00
Doorn, Floris van: maandag 21 Juni om 8:55
Gajic, Dejan: maandag 21 Juni om 9:20
Gregorian, Pablo: diensdag 22 Juni om 11:00
Heijst, Tristan van: diensdag 22 Juni om 15:05
Hooijer, Patrick: maandag 21 Juni om 15:30
Jacobs, Vivian: diensdag 22 Juni om 10:35
Kater, Maarten: diensdag 22 Juni om 9:45
Keijdener, Darius: maandag 21 Juni om 9:45
Kierkels, Arthur: maandag 21 Juni om 8:30
Kluck, Florian: diensdag 22 Juni om 8:55
Lyczak, Julian: diensdag 22 Juni om 9:20
Mulder, Jasper: maandag 21 Juni om 10:10
Mulder, Mark: maandag 21 Juni om 14:15
Muilwijk, Daniel: diensdag 22 Juni om 15:30
Nuiten, Joost: maandag 21 Juni om 10:35
Storm, Reinier: maandag 21 Juni om 11:00
Sybesma, Watse: diensdag 22 Juni om 13:25
Urk, Wester van: diensdag 22 Juni om 13:50
Vákár, Matthijs: maandag 21 Juni om 11:25
Veenhoff, Rudy: diensdag 22 Juni om 14:15
Vorle, Leonardus van de: maandag 21 Juni om 15:55
Vromen, Ludo: maandag 21 Juni om 15:05
Welling, Yvette: diensdag 22 Juni om 14:40
Zwaan, Ian: diensdag 22 Juni om 10:10
The above time indicates when you should show up (you then get the questions, and 20 minutes to prepare). The actual examination starts 25 minutes later.

Eindcijfer = 0,45 × deeltentamen + 0,45 × monderlinge tentamen + 0,1 × inleveropgaves
3143961: 8,9, --, 4,5 3061787: 5,5, --, 3,2 3363120: 7,8, 6,0, 7,5 3230597: 4,4, 5,0, 4,1 0122432: 1,0, --, -- 3379655: 9,6, 9,0, 9,4 3220834: 3,3, --, 3,5 3020215: 5,3, --, 5,5 3345653: 3,3, --, 1,9 3365115: 8,7, 9,5, 8,8 3238784: 5,5, 4,5, 7,0 3242935: 9,1, 6,0, -- 3367428: 8,9, 10,0, 8,9 3220869: 4,0, --, 1,3 0439754: 1,0, --, -- 3173879: 6,2, 7,0, 8,8 3238733: 6,7, 5,0, 2,2 3220672: 6,9, 9,0, 6,3 3061760: 5,3, 3,5, 3,5 3072673: 6,0, 5,5, 6,9 3242927: 3,3, --, 3,1 3233618: 4,4, --, 2,8 3042936: 2,8, --, 3,4 3404994: 8,2, 8,0, 8,0 3375943: 9,1, 6,5, 6,8 3345556: 7,6, 9,5, 9,1 3029298: 7,3, 8,5, 6,4 3029255: 5,8, 6,5, 7,3 3230694: 5,1, --, 2,9 3020959: 6,2, 5,5, 7,5 0123455: 6,7, 9,0, 6,5 3230708: 5,8, 6,0, 6,2 3345602: 8,5, 8,0, 9,2 3034488: 5,1, 7,0, 4,1